Quantum Field Theory II Lectures Notes

Transcription

1 Aprl 3, 06 verson.0 Quantum Feld Theory II Lectures Notes Part I: The Path Integral formulaton of QFT Prof. Dr. Gno Isdor ETH & UZH, Sprng Semester 06

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3 Contents The Path Integral formulaton of Quantum Feld Theory. The Acton Functonal n Classcal Mechancs Path Integral formulaton of QM Path-ntegral formulaton of QM for a genetc Hamltonan Path-Integral formulaton of Scalar Felds PI formulaton of the two-pont correlaton functon The Generatng Functonal Generatng functonal for a free scalar feld Interactng scalar feld theory Path ntegral formulaton of an Abelan Feld Theory Quantzaton of spnor feld Grassmann numbers Path ntegral formulaton for the Drac feld Explct calculaton on functonal determnants Symmetres n Path Integral Ward-Takahash dentty n QED

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5 Chapter The Path Integral formulaton of Quantum Feld Theory. The Acton Functonal n Classcal Mechancs Wthn Classcal Mechancs, the equaton of motons can be derved from a smple prncple: the prncple of Least Acton, or the requrement that the trajectory of the system mnmze a functonal called the Acton. The Acton Functonal can be constructed also for quantum systems and, as we shall see n ths chapter, t allows us to derve a smple brdge between Classcal Mechancs, Quantum Mechancs (QM), and Quantum Feld Theory (QFT). To start, we brefly recall the prncple of Least Acton n Classcal Mechancs. Consder a classcal, non-relatvstc 3-dmensonal system wth a pont-lke partcle whose trajectory s descrbed by the vector x(t) = {x (t), x (t), x 3 (t)}. The equatons of moton reads: mẍ (t) = V x, (.) where V ( x) s the potental energy of the partcle. Solvng the equatons of moton, and makng use of the ntal condtons (poston and velocty) at tme t we can fnd the poston x(t) of the partcle at any tme t. The prncple of Least Acton states that the trajectory of the system can be obtaned by the requrement S[x(t)] = 0, (.) where S s the followng functonal S[ x(t)] = t t dt [E kn E pot ] = t t [ ] dt m x V ( x), (.3) namely the dfference between knetc and potental energy, ntegrated along the trajectory. In order to solve (.), some boundary condtons on the trajectory need to be gven. In ths case t s more convenent to choose them to be the poston of the trajectory at two dfferent tmes (e.g. the ntal and the fnal poston): x(t ) = x (), x(t ) = x (). (.4) In order to prove that (.) leads to the classcal equatons of moton, consder the varaton of S under a small perturbaton x of the trajectory, whch does not affect the endponts

6 coordnates (boundary condtons): S[ x + x] = Integratng by parts we obtan: S[ x + x] = S[ x] + t t t t [ dt m ( x + x) }{{} x + x x ] V ( x + x). (.5) [ dt x m x V ] t ( x) + m dt d ( x t dt x ). (.6) The last term s zero because of the fxed endponts. We thus recover the equatons of moton from the prncple S = 0. Note, however, that the boundary condtons on the trajectory cannot be derved by the prncple and must be gven. The Acton Functonal s also partcularly useful to dentfy the conserved quanttes followng from the nvarance of the system under specfc transformatons (correspondng to symmetres of the system). For nstance, let s assume that the system s nvarant under a rotaton of the coordnates. The nfntesmal transformaton of coordnates (for rotatons along the axs k) s x x + ( α (k) x), ( α (k) x) = α (k) ɛ kj x j, (.7) where α (k) s an nfntesmal parameter and ɛ kj s the completely antsymmetrc tensor (ɛ 3 = ). Because of the symmetry hypothess, we expect α (k) S = 0 under the transformaton (.7). Proceedng ths way we derve the same result as n (.6); however, n ths case the boundary term no longer vanshes. In order to obtan α (k) S = 0 we must mpose ( α (k) x(t ) x(t ) ( ) = α (k) x(t ) x(t ) ). (.8) We thus deduce the exstence of a conserved quantty, whch n ths case s the angular momentum L (k) = mɛ kj x ẋ j. In summary, the Acton Functonal n Classcal Mechancs allow us to ) determne the trajectores of the system once fxed boundary condtons are gven; ) determne the conserved quanttes of the system under specfc symmetry transformatons. As we shall see n the rest of ths chapter, these two propertes hold wth some pecular modfcatons, especally n the case ) also n QM and QFT.. Path Integral formulaton of QM Consder a one-dmensonal quantum system wth the followng Hamltonan Ĥ = p + V (x). (.9) m In the Schrödnger pcture, the probablty ampltude for the state x at tme t = 0 to evolve to the state x at tme t = T s U(x f, x ; T ) = x f e ĤT x, (.0) whch solves the Schrödnger equaton U = ĤU. (.) T

7 Feynman proposed a way to connect U(x f, x ; T ) to the Acton Functonal. In partcular, he proposed the followng expresson U(x f, x ; T ) = Dx(t)e S[x(t)], (.) where the symbol of ntegraton over all paths Dx(t) wll be defned soon. If S[x(t)] the phase of the ntegrand n (.) oscllates very rapdly over the varous trajectores. Ths mples a net vanshng contrbuton to the ntegral but for the statonary trajectory characterzed by x (S[x(t)]) = 0, (.3) xcl that s nothng but the classcal trajectory. The expresson (.) thus allow us to recover n a smple and ntutve way the classcal lmt. We wll now present a proof of the consstency of (.0) wth the prncples of QM n partcular wth the Schrödnger equaton va the dscretzaton of the path. We dvde the tme nterval between 0 and T n n nfntesmal tme slces of duraton t = ɛ. The acton becomes: S = T 0 [ ] n [ dt mẋ V (x) m(x k+ x k ) V ɛ k=0 ( xk+ + x k where x 0 = x and x n = x f. We then defne the (dscretzed) path ntegral by Dx(t) def = N(ɛ) dx N(ɛ) dx N(ɛ) dxn N(ɛ) = N(ɛ) n k= )] ɛ, (.4) dx k N(ɛ), (.5) where N(ɛ) s a normalsaton factor to be determned later. In other words, we ntegrate on all possble values of the ntermedate coordnates x... x n. We want now to demonstrate the consstency of (.0) by nducton method on n, the number of tme slces. To ths purpose, we assume (.0) to be true over the tme nterval (0, T ɛ), and then compute explctly the effect of the last tme slce. Ths mples U(x f, x ; T ) = dx n e S[x] U(x n, x ; T ɛ) = N(ɛ) N(ɛ) dx exp { [ ɛ m(x f x ) V ɛ ( x + x f )]} U(x, x ; T ɛ). Snce the knetc term s rapdly oscllatng unless x f x, we can (Taylor) expand for x close to x f, U(x f, x ; T ) = ( x ) +xf ɛv e [{}}{ dx e m (x f x ) ɛ ] N(ɛ) ɛv (x f) + [ ] + (x x f ) + x f (x x f ) + U(x x f, x ; T ɛ) f }{{} U(x,x ;T ɛ) 3.

8 We can perform the ntegraton over x usng the followng Gaussan ntegraton formulae: π dx e ax = a, dx xe ax = 0, dx x e ax = π a a. (.6) In prncple, these denttes hold only for Re[a] > 0, that s not our case. However, as we shall dscuss later on, we can overcome ths problem addng small dampng terms (small real coeffcents n the exponents) that later on are set to zero. Leavng asde ths techncal pont, we obtan U(x f, x ; T ) = [ N(ɛ) ] [ π ɛ ɛ m V (x f) + ɛ m x f + O(ɛ ) ] U(x f, x ; T ɛ). (.7) In order to gve a meanng to the lmt ɛ 0 we deduce that we need to mpose the condton [ ] π ɛ =. (.8) N(ɛ) m We are now able to take the lmt ɛ 0, that yelds [ T U(x f, x ; T ) = m ] + V (x f ) U(x f, x ; T ) = ĤU(x f, x ; T ). (.9) x f As can be seen, we recover the Schrödnger equaton, demonstratng the consstency of the Feynman Path-Integral formulaton of QM n ths smple system..3 Path-ntegral formulaton of QM for a genetc Hamltonan Consder now a generc system wth N coordnates q = {q } and a generc Hamltonan H ( q, p). The transton ampltude from the ntal state q I to the fnal state q F s gven by ĤT q F e q I. (.0) Insertng the dentty I = dq k qk q k n each tme slce, the transton ampltude can be expressed by the product of terms of the type ( q n q F, q q I ) ( ) Ĥɛ q k+ e qk = q k+ I Ĥɛ + Let s frst assume that H has the followng separable form a) Evaluaton of the contrbuton from ˆf( q) q k. (.) Ĥ = ˆf( q) + ĝ( p). (.) ( ) q k+ ˆf( q) qk+ + q k ( ) q k = f (qk+ q qk+ + q k k) = f dp p k k ( q k+ q k ) π e, (.3) The hat on the varous terms s there to remnd us that these are operators. 4

9 where we have used ˆf( q) q = f( q) q and dp π epq = (q). b) Evaluaton of the contrbuton from ĝ( p) q k+ ĝ( p) q k = q k+ π dp kg( p k ) p k p k q k, (.4) where we have ntroduced momentum egenstates va the nserton of I = dp p p. π Usng also p k q k = e q k p k and q k p k = e + p k q k we get q k+ g( p k ) q k = = π π dp k q k+ p k g( p k )e q k p k dp kg( p k )e p k ( q k+ q k ). (.5) Takng the full Hamltonan, we fnally get q k+ Ĥ q k = ( ) dp k π H qk + q k+, p k e p k ( q k+ q k ). (.6) Note that on the rght hand sde H s just a functon of the varables, whereas of the left hand sde t s an operator. For generc Ĥ we have a potental problem snce p and q don t commute: we need to specfy ther order. To ths purpose, t s convenent to defne the so-called Weyl orderng of Ĥ. Ths s defned by (.6), namely Ĥ s Weyl ordered f ( ) q k+ Ĥ q qk + q k+ k d p k H, p k e p k ( q k+ q k ). (.7) It s then easy to check that, for nstance, Ĥ = p q s not Weyl ordered, whle Ĥ = 4 ( p q + q p + qp q ) (.8) t s Weyl ordered. If Ĥ s not Weyl ordered, t can always be put n a Weyl-ordered form usng the commutaton relatons between q and p. Ths way we can contnue our dscusson n full generalty, relaxng the assumpton of a separable structure for Ĥ, as n Eq. (.), but smply requrng Ĥ s Weyl ordered. We can now go back to the sngle tme slce term n (.): q k+ e Ĥɛ qk = dp k π e = dp k π e+ ( ɛ H qk + q k+, p k )e p k ( q k+ q k ) [ ɛ p k q k+ q k ɛ (.9) ( qk )] + q H k+, p k. (.30) The underlned term has the followng lmt n the contnuum (ɛ 0): [ p k q ( )] k+ q k qk + q [ k+ H, p k p k ɛ q ] k H ( q k, p k ). (.3) 5

10 Then takng all tme slces nto account, we fnally obtan q F e ĤT qi = dq k dp ɛ k π e k[ p k q k H(q k, p k )] (.3) k { T [ = DqDp exp dt p q H ( q, p)] } (.33) 0 where the ntegral over the trajectores q(t) s constraned at t = 0 and t = T, whle the one on p(t) s free. The expresson n (.3) can be nterpreted as the formal defnton of the path ntegral for generc QM systems. As can be seen, the formulaton s qute smple and elegant. It s also worth to notce that n ths case the measure of the ntegral s the standard quantum phase-space measure..4 Path-Integral formulaton of Scalar Felds The Path Integral (PI) formulaton dscussed n the prevous secton, vald for a system wth N coordnates, s well suted for a generalzaton to the case of a scalar feld theory. To ths purpose, we need to generalze the result to a system wth an nfnte number of coordnates: one quantum coordnate for each pont of the 3-dmensonal space: q k (t) φ( x, t), p k (t) Π( x, t) L φ( x, t), (.34) wth the followng Hamltonan: Ĥ = [ d 3 x Π + ] ( φ) + V (φ). (.35) From now on, except when explctly stated, we set =. Generalzng the result n (.33), the transton ampltude from the ntal state confguraton φ I ( x) at tme t = T to the fnal state confguraton φ F ( x) at tme t = +T s Some comments are n order: φ F (x) e Ĥ(t F t I ) φ I (x) = Π=Π φ Dφe + +T T d4 x[ ( µφ) V (φ)] = DφDΠe + +T T d4 x[π φ Π ( φ) V (φ)] (.36) Dφe +T T d4 xl(φ, µφ) (.37) In gong from (.36) to (.37) we have performed a Gaussan ntegral over the shft momentum varable Π. Ths ntegral leads to an overall constant (φ-ndependent) factor that we can re-absorb n the defnton of Dφ. The ntegral extends over all the the feld confguratons connectng the two boundary condtons at t I = T and t F = +T, namely φ F ( x, +T ) and φ I ( x, T ). The result n manfestly Lorentz nvarant except for these boundary condtons. From now on we adopt (.37) as the defnton of the PI formulaton of a Scalar Feld Theory. Ths mples we wll characterze the dfferent feld theores n terms of the Lagrangan of the system rather than the Hamltonan. 6

11 .4. PI formulaton of the two-pont correlaton functon A further step toward a more convenent formulaton of QFT s obtaned by gettng rd of the boundary condtons. What we are really nterested n, s the set of tme-ordered correlaton functons among felds at dfferent space-tme ponts, evaluated on the ground state of the system (also know as Green s functons). Indeed, as we know from QFT-, startng from these correlaton functons the LSZ reducton formulae allow us to extract the probablty ampltudes for physcal scatterng processes,.e. the elements of the S matrx. The smplest of such correlatons s the two-pont functon Ω T { ˆφ H (x ) ˆφ H (x )} Ω, (.38) where Ω denotes the ground state (of the nteractng theory) and ˆφ H the operators assocated to the felds n the Hesenberg s pcture. In order to fnd a PI formulaton of (.38), consder the followng functonal ntegral, wth generc boundary condtons at t = ±T I = Dφ(x)φ(x )φ(x )e +T T d4xl(φ). (.39) The man functonal ntegral Dφ(x) can be broken up as Dφ(x) Dφ ( x)dφ ( x) Dφ(x), (.40) φ(x 0, x) = φ ( x) φ(x 0, x) = φ ( x) namely ntroducng two specfc boundary condtons at tmes x 0 and x 0 (n addton to those at the endponts T and T ), but ntegratng separately over the correspondng fxed-tme feld confguratons φ ( x) and φ ( x). Consder frst the case T > x 0 > x 0 > T. Under ths tme orderng, the exponental term n (.39) can be decomposed as 0 x 0 e T T d4xl(φ) = e x 0 T d4xl(φ) e x d 4 xl(φ) T e x 0 d 4 xl(φ). (.4) After the decomposton of the man ntegral, the extra factors φ(x ) and φ(x ) n (.39) become φ ( x ) and φ ( x ), and can be taken outsde the nner ntegral. We thus obtan I (x > x ) = Dφ ( x)dφ ( x)φ ( x )φ ( x ) (.4) φ F e H(T x0 ) φ φ e H(x0 x0 ) φ φ e H(x0 +T ) φ I }{{}}{{}}{{} T Dφe x 0 d 4 xl(φ) x0 Dφe x 0 d 4 xl(φ) Dφe x0 T d4 xl(φ). The factors φ ( x ) can be transformed nto nsertons of Schrödnger operators, by usng ˆφ S (x 0, x ) Dφ ( x)φ ( x ) φ φ, (.43) leadng to I (x > x ) = φ F e H(T x0 ) ˆφS (x 0, x )e H(x0 x0 ) ˆφS (x 0, x )e H(x0 +T ) φ I (.44) = φ F e HT ˆφH (x ) ˆφ H (x )e HT φ I. (.45) 7

12 Puttng together the case x > x and x > x we arrve to I = Dφ(x)φ(x )φ(x )e +T T d4xl = φ F e HT T { ˆφ H (x ) ˆφ H (x )}e HT φ I. (.46) We thus obtaned the desred tme-ordered product, but evaluated on the generc endpont feld confguratons, rather than on the ground state of the system. Consder now the rato I /I 0, where I 0 = Dφ(x)e +T T d4xl = φ F e H(T ) φ I, (.47) and consder further the lmt T + ( ɛ), where ɛ s an arbtrarly small real parameter. In order to evaluate ths lmt, we can project the ntal state φ I nto the complete sum of the egenstates n of the Hamltonan. For T + ( ɛ), the acton of e HT selects only the ground state component whch s the least suppressed(the addtonal terms beng exponentally suppressed): e HT φ I = n e HT n n φ I T + ( ɛ) e E 0T Ω Ω φ I. (.48) A smlar procedure can be appled to the the fnal state φ F. As a result, we obtan I = e E 0T φ F Ω Ω φ I Ω T {φ(x )φ(x )} Ω Ω T {φ(x )φ(x )} Ω. (.49) The proportonalty constant on the r.h.s. of (.49) drops out n the rato I /I 0, leadng to Ω T {φ(x )φ(x )} Ω = Dφ(x)φ(x )φ(x )e S Dφ(x)e S (.50) where S = d 4 L(x). N.B.: The rato n (.50) s the reason why we can often neglect the overall normalzaton of the functonal ntegral..5 The Generatng Functonal We now need a method able to generate n a systematc way all the relevant correlaton functons, namely all the correlaton functons nvolvng an arbtrary number of felds at dfferent space-tme ponts. As we shall show, all these correlaton functons are encoded n one man transton ampltude: the groundstate-to-groundstate transton ampltude n presence of arbtrary drvng force J(x), or the Generatng Functonal. The basc physcal pcture behnd the Generatng Functonal s the followng: by means of an arbtrary source J(x) we create an arbtrary exctaton of ntermedate states that, n turn, gve us access to all possble correlaton functons. Techncal tool: functonal dervatves. Smlarly to the Acton, S[φ], that s a functonal of φ, the Generatng Functonal, W[J], s a functonal of J. In order to extract the correlaton From now on we omt the specfc subscrpts to denote the Hesenberg pcture: ˆφH (x) φ(x). 8

13 functons from W [J] we need to ntroduce the techncal tool of functonal dervatves. The man rules of functonal dervatves are lsted below J(x) J(y) = 4 (x y), d 4 yj(y)f(y) = f(x), (.5) J(x) J(x) F [J(y)] = F [J 4 (x y), separately for each J]. (.5) Smlarly to ordnary dervatves, t follows that d J(x) e 4yJ(y)φ(y) = φ(x)e d 4yJ(y)φ(y), (.53) J(x) d 4 y µ J(y)φ(y) P.I. = µ φ(x). (.54) We are now ready to go back to W [J]. The Generatng Functonal for a scalar feld theory s defned by W [J] = Ω Ω J = Dφe d 4x[L(φ)+J(x)φ(x)], (.55) where J(x) s an arbtrary functon. Startng from ths defnton, t s easy to verfy that ( ) ( ) W [J] = Dφ(x)φ(x )φ(x )e d 4x[L(φ)+J(x)φ(x)]. (.56) J(x ) J(x ) As a result, the two-pont correlaton functon can be wrtten as Ω T {φ(x )φ(x )} Ω = ( ) ( ) W [J] W [0] J(x ) J(x ). (.57) J=0 Ths result can be easly generalsed to arbtrary correlaton functons. We can then express W [J] be means of a (functonal) Taylor expanson as n W [J] = W [0] dx dx n J(x ) J(x n )G (n) (x,, x n ), (.58) n! n=0 where the coeffcent of the expanson, namely the functons G (n) (x,, x n ), are the n-pont Green s functons G (n) n ( ) (x,, x n ) = W [J] W [0] J(x ) = J=0 = Ω T {φ(x )φ(x )... φ(x n )} Ω. (.59).5. Generatng functonal for a free scalar feld For a free scalar feld, the argument at the exponent of the generatng functonal s [ S J = d 4 x ( µφ)( µ φ) ] m φ + Jφ [ ɛ d 4 x ( µφ)( µ φ) ] (m ɛφ ) + Jφ { = d 4 x φ [ m + ɛ ] } φ + Jφ (.60) 9

14 where the ɛ term has been ntroduced n order to make the ntegral convergent, and the last dentty has been obtaned through ntegraton by parts. In (.60) we have obtaned a quadratc form n φ that can be ntegrated explctly ( Gaussan ntegral). To ths purpose, we need to fnd the nverse of the operator ( m +ɛ), or the functon D(x y) defned by As known from QFT-, ths functon s D(x y) = ( m + ɛ)d(x y) = 4 (x y). (.6) d d k (π) 4 k m + ɛ e k (x y). (.6) In a formal sense, that acqure a well-defned meanng nsde functonal ntegrals by means of (.6), we can wrte D(x y) = ( m + ɛ). (.63) Ths allows us to complete the square n (.60) va the shft φ(x) φ (x) = φ d 4 yd(x y)j(y) = φ(x) + ( m + ɛ) J(x), (.64) obtanng S J = [ ] d 4 x φ ( m + ɛ)φ d 4 xd 4 yj(x)d(x y)j(y). (.65) The change of varable doesn t affect the measure, snce the Jacoban of the transformaton s one. The ntegral on Dφ s not well defned, but ths only leads to an rrelevant (J-ndependent) normalzaton factor. We thus obtan the followng smple expresson for W [J] n the free feld case 3 W 0 [J] = W 0 [0]e d 4 xd 4 yj(x)d(x y)j(y) = W0 [0]e JxDxyJy, (.66) where the second dentty s nothng but a convenent compact notaton. As far as the normalzaton W 0 [0] s concerned, ts formal expresson s gven by d 4 xφ [ m + ɛ]φ W 0 [0] = Dφ e. (.67) By analogy wth the fnte-dmensonal Gaussan ntegrals, dx k e x B j x j (det B), (.68) we can express W 0 [0] as k W 0 [0] [det( m + ɛ)]. (.69) Ths result would be well defned f we were able to prove that the determnant s a postve quantty and t converges (the number of egenvalues could be nfnte). Ths can be acheved 3 The subscrpt 0 n W 0 [J] denotes the free-feld case. 0

15 assumng approprate boundary condtons on the system. However, as antcpated, for most of the applcatons we are nterested n, the explct expresson of W [0] s rrelevant. Usng the explct expresson of W 0 [J] we can now compute some correlaton functons (n the free-feld case). For the two-pont functon we fnd Ω T φ(x )φ(x ) Ω = J = J e JxDxyJy J J=0 [ D yj y ] D xj x e J=0 JxDxyJy = D + D = D = D(x x ), (.70).e. we recover the expresson of the free scalar propagator. obtan For the four-pont functon we Ω T φ φ φ 3 φ 4 Ω = [ J x D x4 ]e J J J 3 = [D 34 + J x D x4 J y D y3 ]e J J = JxDxyJy J=0 JxDxyJy J=0 [D 34 J x D x + D 4 J y D y3 + J x D x4 D 3 ]e J JxDxyJy J=0 = D 34 D + D 4 D 3 + D 4 D 3, (.7) namely the sum of three terms, each contanng two dsconnected contrbutons. Besdes W [J], we can ntroduce also the functonal Z[J], defned by W [J] = e Z[J], (.7) that n the free-feld case assumes the form Z 0 [J] = J xd xy J y + const. (.73) The Taylor decomposton of Z[J] can be wrtten n terms of new set of Green functons: Z[J] = n=0 n n! dx dx n J(x ) J(x n )G (n) C (x,, x n ). (.74) In the free-feld case we fnd G () c = D(x y), G (n>) c = 0. (.75) As we wll show n the next paragraph, by means of the explct calculaton n the case of the nteractng φ 4 theory, the G C functons are nothng but the the connected Green functons of the theory.

16 .5. Interactng scalar feld theory We wll consder now an nteractng Lagrangan of the form We have just seen that L = L 0 (φ) V (φ). (.76) d J(y) e 4xJ(x)φ(x) = φ(y)e d 4xJ(x)φ(x), (.77) then for a generc (analytc) functonal F (φ) we have ( F (φ)e d 4xJφ = F ) e d 4 xjφ J and thus W [J] = Dφe d 4 xv ( Jx ) e d 4 x[l 0 +Jφ] = e d 4 xv ( Jx ) e JxDxyJy W 0 [0] }{{} W 0 [J] (.78). (.79) We now want to analyse the case n whch V can be trated as a small perturbaton. In ths case we can perform a perturbatve expanson of e V around. We can thus wrte W [J] as W [J] = W 0 [J] { + W 0 [J] [e d 4 xv ( ) ] } Jx W 0 [J] (.80) and recallng that W [J] = e Z[J] we fnd { Z[J] = Z 0 [J] + ln + e Z 0[J] [e d 4 V ( ) ] } Jx e Z 0[J] + const., (.8) where we have defned W 0 [J] = W 0 [0] ez 0[J] = e JxDxyJy. (.8) Let s now apply these generc formulae to the specfc example, namely the case of the potental For small enough λ we get e d 4 xv ( Jx ) λ 4! and the logarthm n (.8) becomes ln { + } = λ 4! e Z 0[J] where = = = V = λ 4! φ4. (.83) d 4 z( ) 4 4 J(z) 4 + O(λ ) (.84) 4 d 4 z J(z) 4 ez 0[J] + O(λ ) (.85) d 4 4 z J(z) 4 e JxDxyJy = d 4 3 ( ) z D J(z) 3 xz J x e JxDxyJy d 4 [ ] z ( D J(z) zz + D xz J x D yz J y )e JxDxyJy d 4 ] z [(D zz D yz J y + D zz D yz J y D xz J x D yz J y D αz J α )e JxDxyJy J(z) d 4 z [ ] 3Dzz 6D zz D yz D xz J y J x + D xz D yz D αz D bz J x J y J α J b e JxDxyJy (.86)

17 As antcpated, we have only connected Green s functons, that n the last lne we have convenently expressed by means of Feynman dagrams. The fnal result for the functonal F [J] s then Z[J] = C J xd xy J y λ 4! [3D zz 6D zz D yz D xz J y J x Usng ths result n Eq. (.8) we can get back to W [J] obtanng + D xz D yz D αz D bz J x J y J α J b ] + O(λ ). (.87) Snce vacuum bubbles always cancel n the correlaton functons, for smplcty, t s absorbed nto W 0 [0], whch becomes W [0]. Computng G(x, x, x 3, x 4 ) we get the same results as n the free case plus the nserton of the nteractng terms at O(λ)..6 Path ntegral formulaton of an Abelan Feld Theory We now proceed extendng the path-ntegral formulaton of QFT to theores wth dfferent type of felds (and correspondngly dfferent free actons). In ths secton we consder the case of Abelan gauge felds, namely the massless vector felds (A µ ) appearng n the Maxwell Lagrangan: L = 4 F µνf µν, F µν = µ A ν ν A µ. (.88) By constructon, the generatng functonal for ths feld s W [J µ ] = DA e {S[A]+ J µa µ } (.89) wth [ S[A] = d 4 x ] 4 ( µa ν ν A µ )( µ A ν ν A µ ) = d 4 x [ µ A ν µ A ν ν A µ µ A ν ] = d 4 xa µ [ g µν µ ν ]A ν, (.90) 3

18 where the last relaton follows from ntegraton by parts. To proceed as n the case of the scalar theory we need to dentfy the free-feld propagator, or the functon D µν (x y) solvng the followng equaton ( g µν µ ν )D νρ (x y) = g ρ µ 4 (x y) (.9) ( k g µν + k µ k ν + ɛ) D νρ (k) = g ρ µ (n momentum space). (.9) However, t turns out that we cannot proceed snce ( k g µν + k µ k ν ) s a sngular (or not nvertble) operator, gven that Ths fact s a consequence of the gauge nvarance of the acton: ( k g µν + k µ k ν )k ν = 0. (.93) S[A] = S[A ] for A µ = A µ + e µα(x), (.94) that, n turn, mples that n the path-ntegral (.89) we are redundantly ntegratng over a contnuous nfnty of physcally equvalent feld confguratons. To fx the problem, we would need to solate the nterestng part of the functonal ntegral, countng each physcal confguraton only once. In order to select a specfc gauge we ntroduce a gauge-fxng condton, such as µ A µ = ω(x). (.95) Defnng the functon G(A) = µ A µ ω(x), the above gauge-fxng condton s equvalent to mpose G(A) = 0. Let s now consder the gauge-transformed feld A α µ(x) = A µ (x) + e µα(x), (.96) G(A α ) = µ A µ + e α ω(x). (.97) Generalzng the followng dentty ( = ) ( ) da (n) g (g(a,, a n )) det a j (.98) to functonal ntegrals we can wrte I = select gauge confguraton {}}{ DG(A α ) [ G(A α ) }{{} gauge fxng condton ] = ( ) G(A α ) Dα(x) det [G(A α )]. (.99) α }{{} det( e ) The last expresson s the key relaton that allows us to splt n the functonal ntegral (.89) the redundant part from the non-trval ntegraton over physcal nequvalent feld confguratons. 4

19 Indeed nsertng the above dentty n the generatng functonal we can wrte ( ) G(A W [J µ ] = DA Dα [G(A α α ) )] det e S[A] e d 4 x J A α }{{} C = C DA α Dα [G(A α )]e S[Aα] e d 4 x J A α A α A = C Dα DA [G(A)]e S[A] e d 4 x J A }{{} C = C DA [G(A)]e S[A] e d 4 x J A (.00) The overall factor C s badly dvergent, but ths s not a problem snce we can reabsorb ths dvergence n the defnton of the measure n the DA space and, as we have already seen, the overall factor n functonal ntegral drops out n physcal correlaton functons. Summarzng, we can obtan Eq. (.00) startng from Eq. (.89) thanks to the followng key observatons: S[A] = S[A α ] d 4 xj µ A µ = d 4 xj µ A µ α f J µ s a conserved current DA ( = DA α ) [the measure s unchanged under gauge transormatons] det G(A α ) = det ( ) s ndependent of A. α e As we wll dscuss later on n ths course, the last pont does not hold for non-abelan gauge theores. Let s go back to the result n Eq. (.00). We would lke to further transform t n order to obtan a smple Gaussan ntegral (as n the case of the scalar feld). To ths purpose, we note that the choce of ω(x) s arbtrary. We can thus consder a properly weghted combnaton of dfferent ω(x) functon. In partcular, we can consder the followng combnaton W [J µ ] = C (ξ) Dω e d 4 x ω ξ DA ( µ A µ ω)e S[A] e d 4 x J A }{{} Gaussan ntegral centered n ω = 0, wth varance ξ = C (ξ) DA e S[A] e d 4 x ( µa µ ) ξ e d 4 x J(x)A(x) { [ = C (ξ) DA exp d 4 x ]} 4 F µνf µν ξ ( A) e d 4 x J(x)A(x) }{{} effectve term n the Lagrangan = C (ξ) { DA exp d 4 x A µ [ g µν µ ν + ξ µ ν ] }{{} ths operator can now be nverted A ν }e d 4 x J(x)A(x). (.0) Ths way we have fnally reached an nvertble quadratc form n the exponent. Ths allows us to obtan a well-behaved propagator, D νρ (x y), defned by 4, ( [( ɛ)g µν µ ν )] D νρ (x y) = g ρ µ 4 (x y) (.0) ξ 4 The ɛ term s added to make the ntegral convergent. 5

20 or, n the momentum space, ( [( k ɛ)g µν + k µ k ν )] D νρ (k) = g ρ µ. (.03) ξ Usng the general ansatz t s easy to fnd that D νρ (k) = D νρ (k) = Ag νρ + Bk ν k ρ, (.04) ] [g νρ kν k ρ ( ξ) k + ɛ k. (.05) For obvous reasons the parameter ξ s denoted gauge-fxng parameter. choce for t, that allows to get partcularly smple propagators, are Two notable ξ = 0 [Landau gauge] and ξ = [Feynman gauge]. (.06) The ξ-dependence should drop n any correlaton functon nvolvng gauge-nvarant operators, O(x), namely 0 T {O(x ) O(x n )} 0 = DA O(x ) O(x n )e S ξ[a] DA e S ξ [A], (.07) where S ξ [A] = [ d 4 x 4 F µνf µν ]. ξ.7 Quantzaton of spnor feld.7. Grassmann numbers In order to generalze the path-ntegral method to spnor felds we need to ntroduce a new type of varables: the Grassmann numbers. The defnng property of two Grassmann numbers θ and η s {θ, η} = 0, θη = ηθ = θ = η = 0 (.08) In ths secton we wll use latn letters for ordnary numbers and greek letters for Grassmann numbers. Any functon f(θ) can be expanded to the frst order as f(θ) = { a + βθ = a θβ a + bθ = a + θb (.09) wth β a Grassmann varable and b an ordnary number. We also defne the dervatve wth respect to a Grassmann varable by the relaton { } d dθ, θ =, (.0) such that d dθ f(θ) = { d (βθ) = d (θβ) = β dθ dθ d (bθ) = b. (.) dθ 6

21 The ntegral over a Grassmann varable s defned by dθ = 0, dθθ =, (.) such that dθf(θ) = dθ(βθ) = dθ(bθ) = b dθ(θβ) = β (.3) Ths mples dθf(θ) = dθf(θ + η), (.4) namely the nvarance of the ntegral under a shft of varables. It s also useful to defne complex Grassmann varables, θ = θ + θ, θ = θ θ, (.5) wth θ and θ ordnary (real) Grassmann varables. It s easy to check that θθ = (θ + θ )(θ θ ) = ( θ θ + θ θ ) = θ θ (.6) θ θ = (θ θ )(θ + θ ) = (θ θ θ θ ) = +θ θ (.7) that mples that θ and θ can be treated as ndependent (Grassmann) varables: {θ, θ } = 0. Fnally we defne (θη) = η θ = θ η (.8) We are now ready to evaluate Gaussan-type ntegrals over complex Grassmann varables. The smplest example s dθ dθe bθ θ = dθ dθ[ bθ θ] = b dθ dθθ θ = b dθ θ dθdθ = b. (.9) Ths has to be compared wth the analog ntegeral over ordnary complex varables, dz dze bz z = π b. (.0) Smlarly dθ dθ(θθ )e bθ θ = dθ dθ(θθ )[ bθ θ] = dθ dθ(θθ ) = = b b, (.) can be compared wth dz dz(z z)e bz z = π b = π b b. (.) Consderng a generc Hermtan matrx B j wth non-vanshng egenvalues b, and generalzng the above results, one obtans dθ dθ e θ B jθ j = dθ dθ e θ b θ = det(b). (.3) dθ dθ θ k θl e θ B jθ j = det(b)(b ) kl. (.4) 7

22 .7. Path ntegral formulaton for the Drac feld We defne the Grassmann feld ψ(x) as ψ(x) = ψ φ (x), (.5) where ψ are Grassmann numbers and φ (x) s an ordnary functon of x. To descrbe the Drac feld, we dentfy the φ wth the components of a (4-component) Drac spnor. We can then to ntroduce the generatng functonal W [ η, η] = D ψdψe d 4 x [ ψ( / m)ψ + ηψ + ψη ], (.6) where η and η are external Grassmannan sources. Proceedng as n the case of scalar felds, we extract the free-feld propagator by solvng and obtanng (/ m + ɛ)d F (x y) = 4 (x y), (.7) D F (x y) = d 4 k /k m + ɛ e k(x y). (.8) Then by a shft of varables n the functonal ntegral we can re-wrte the generatng functonal as W [ η, η] = W 0 e d 4 xd 4 y η(x)d(x y)η(y). (.9) It s straghtforward to check that, as expected, 0 T {ψ(x ) ψ(x )} 0 = ( ) ( ) + W [η, η] W 0 η(x) η(x) η= η=0 D ψdψψ(x ) ψ(x )e S = D ψdψe. (.30) S Havng ntroduced a path-ntegral formulaton for Drac and Abelan gauge felds, t s easy to derve the expresson of the Generatng functonal of QED. By constructon the latter s [ W [ η, η, J µ ] = N D ψdψda d 4 x µ e 4 F µνf µν ] ξ ( µa µ ) + ψ( /D m)ψ e d 4 x [ ηψ + ψη + A µ J µ] (.3) where D µ = µ + ea µ. Treatng the nteracton term n the acton, namely L = e ψ(x)γ µ ψ(x)a µ (x) (.3) as a perturbaton, and proceedng as n Sect..5., one recovers the Feynman rules of QED derved usng the formulaton of QFT dscussed n QFT- (canoncal quantzaton). 8

23 .7.3 Explct calculaton on functonal determnants As outlned above and shown explctly n Sect..5., the path-ntegral formulaton of QFT allows us to recover known results obtaned by means of canoncal quantzaton. However, the path-ntegral formulaton s more general and, n some cases, allow to derve n a smple and transparent way results that are not easy to obtan by means of the canoncal-quantzaton formalsm. Two examples of ths statement are show n ths and n the followng secton. As a frst example, we present here the explct calculaton of the functonal ntegral for the Drac feld n presence of an external vector feld (that could possbly by dentfed wth the electromagnetc feld n the lmt where we treat the latter as a non-dynamcal background feld). The generatng functonal we are nterested n s W [J µ ] = D ψdψ e d 4 x [ ψ( / m)ψ ej µ ψγ µ ψ ] { [ ]} det( /D J m) = det (/ m) / m ( e /J) [ ] = W 0 det / m ( e /J). (.33) To evaluate explctly ths functonal determnant we frst note that, for a generc matrx A wth egenvalues a, we can wrte det(a) = a = e log(a ) = e log(a ) = e Tr[log A]. (.34) Then defnng W [J µ ] = e Z[Jµ], and applyng the above decomposton to the functonal determnant n Eq.(.33) we obtan the followng formal expresson for Z[J µ ] { [ ]} Z[J µ ] = Z 0 Tr log / m ( e /J) ( = Z 0 ) {[ ] n } Tr n / m ( e /J) (.35) n= }{{} C n In order to nterpret the meanng of ths expresson t s convenent to consder the lmt of dscretzed and fnte space tme. In ths case the varous (functonal) operators we are consderng can be dentfed as matrces leavng n a vector spaces labeled by the (fnte) number of space-tme ponts [e.g.: D(x y) D x y j = D(x x j )]. Wth ths dentfcaton s easy to realze that Tr = dx (.36) x,r r where r denotes the spnor ndces. Recallng also that the nverse of the operator (/ x m) s nothng but the propagator of the Drac feld, we obtan the followng expressons: C = ( )( e) d 4 x tr[d(x x)/j(x)] (.37) C = ( e) d 4 xd 4 y tr[d(x y)/j(y)d(y x)/j(x)] (.38) C n = n ( e)n d 4 x d 4 x n tr[d(x x )/J(x ) D(x n x )/J(x )] (.39) 9

24 where now the trace refers only to spnor ndces. Dagrammatcally, the above result corresponds to As expected, the functonal Z[J] contans only connected Green functons. Exponentatng Z[J] we obtan the complete functonal ntegral wth connected and dsconnected dagram. The calculaton of ths functonal determnant was partcularly smple due to non-nteractng nature of the external feld. Stll, t served to the purpose of llustratng the smplcty and the powerfulness of ths method..8 Symmetres n Path Integral Another aspect where the path ntegral formulaton of QFT s partcularly useful s n the explotaton of the symmetres of the theory. Let s start consderng an example, namely the global Abelan symmetry of a complex scalar feld theory. The Lagrangan of the system s and s nvarant under the global U() transformaton L = µ φ µ φ m φ φ (.40) φ(x) e α φ(x) φ(x) + αφ(x). (.4) At the classcal level, thanks to Noether s theorem, we know that ths nvarance mples the exstence of a conserved current ( µ J µ = 0). In ths specfc example, the conserved current s J µ = L ( µ φ) φ + L ( µ φ ) ( φ ) = ( µ φ φ µ φφ ). (.4) What happens at the quantum level? The dervaton of Noether s theorem reles on the assumpton that the system obeys the classcal equatons of motons. Ths fact no longer holds at the quantum level: n the functonal ntegral we consder also trajectores that do not mnmze the acton. It s therefore not obvous what happens to the relaton µ J µ = 0 when we consder a quantzed system. In order to nvestgate what happens at the quantum level, let s consder the followng two-pont correlaton functon, 0 T {φ(x )φ (x )} 0 = N Dφφ(x )φ (x )e S[φ]. (.43) 0

25 Let s then apply a change of varables (n the felds) correspondng to a local U() transformaton, namely φ(x) φ (x) = e α(x) φ(x) φ(x) + α(x)φ(x) (.44) The measure Dφ s unchanged, snce the change of varables s a untary transformaton, we therefore fnd Dφφ(x )φ (x )e S[φ] = Dφ φ (x )φ (x )e S[φ ] = Dφφ (x )φ (x )e S[φ ]. (.45) Consderng the frst and the last term n ths dentty, and expandng the latter to frst order n α we obtan [ 0 = Dφ α(x )φ(x )φ (x ) α(x )φ(x )φ (x ) + φ(x )φ(x ) d 4 x S ] α α(x) e S[φ] (.46) where S α = α(x) = α(x) = α(x) d 4 yl(φ + αφ) [ d 4 y L(φ) + L φ φ + L φ φ }{{} =α(y)φφ α(y)φφ =0 + L ( µ φ) ( µα)φ L ] ( µ φ ) ( µα)φ d 4 y µ α(y) ( µ φ φ µ φφ ) }{{}. (.47) J µ Gven the acton s nvarant under global transformatons, the varaton of the acton under the local transformaton s necessarly proportonal to µ α, and we found that the proportonalty factor s nothng but the current J µ (.e. the current that s conserved at the classcal level). Note that, contrary to Noether s theorem, to obtan ths result we have not used the fact that the feld satsfy the equatons of moton. Then ntegratng by parts we fnally obtan S α = µj µ (x). (.48) Usng ths result n Eq. (.46) we get [ ] 0 = Dφ α(x )φ(x )φ (x ) α(x )φ(x )φ (x ) φ(x )φ(x ) d 4 xα(x) µ J µ (x) e S[φ] = d 4 xα(x) Dφ[(x x )φ(x )φ (x ) (x x )φ(x )φ (x ) φ(x )φ(x ) µ J µ (x)]e S[φ]. (.49) Snce the above result must be vald for any α(x), we fnally obtan the followng dentty 0 T { µ J µ (x)φ(x )φ(x )} 0 = 0 T {φ(x )(x x )φ (x )} 0 0 T {φ(x )(x x )φ (x )} 0. (.50) Ths result s a representatve example of the modfcaton of the concept of conserved current n QFT: the current s conserved up to contact terms. In the canoncal-quantzaton formalsm the later appear due to the non-trval commutaton relatons among the felds. In the

26 path-ntegral formulaton these contact terms smply arse by the defnton of the correlaton functons, as llustrated above. It should be clear that the above result can be generalzed to generc n-pont functons, and also to more complex sets of symmetry transformatons. The correspondng relatons are known as Ward-Takahash denttes (for contnuos global transformatons) or, more generally, Schwnger-Dyson equatons. The general expresson of the latter, obtaned proceedng as n the example above but for: ) startng from a generc n-pont correlaton functon; ) consderng a generc (untary) change of varables n the felds s n 0 T { (x)φ(x ) φ(x n )} 0 = 0 T { (x)φ(x ) ( (x x )) φ(x n )} 0 (.5) = where (x) = S[φ] φ(x) = L ( ) L φ µ. (.5) ( µ φ) Ths result can be consdered the quantum verson of Euler-Lagrangan equatons..8. Ward-Takahash dentty n QED A notable applcaton of the Ward-Takahash dscussed above s the connecton between 3- and -pont functons n QED, that leads to the relaton Z = Z derved n perturbaton theory at O(e ) n QFT-. The relevant (global) symmetry s ψ(x) ψ (x) = e α ψ(x), (.53) leadng to the conserved current J µ e.m. = ψγ µ ψ. Proceedng as n the prevous secton (takng nto account the varous sgns due to the Drac algebra), we get x µ 0 T {J µ (x) ψ(x )ψ(x )} 0 = 0 T {ψ(x ) ψ(x )} 0 (x x ) If we then perform a Fourer transformaton, va d 4 xe qx d 4 x e p x + 0 T {ψ(x ) ψ(x )} 0 (x x ). (.54) d 4 x e p x, (.55) we obtan the followng dentty among correlaton functon n momentum space q µ M µ (q; p, p ) = M(p q, p ) + M(p, q + p ). (.56) or, usng QFT- notaton (see Peskn and Schroeder), q µ Γ µ (p, p ) = Σ(p ) Σ(q + p ), (.57) Here Γ µ (p, p) s the vertex functon descrbng the nteracton of the electron wth the photon feld and Σ(p) s the one-partcle rreducble correcton to the electron propagator. Expandng these correlaton functons for p, m and q 0 Σ(p) = Σ(p) p =m + Z (/p m) +... Γ µ (p, p) = Z γ µ +... (.58) we fnally get the relaton Z = Z. Note that, whle ths relaton was derved by means of an explct calculaton at O(e ) n QFT-, we have now derved t n a general way that s vald to all orders n perturbaton theory.