Noether Theorem, Noether Charge and All That

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1 Noethe Theoem, Noethe Chage and All That Ceated fo PF by Samalkhaiat 10 Tanfomation Let G be a Lie goup whoe action on Minkowki pace-time fomally ealized by coodinate tanfomation ( ) ( 1,3,η) M i Infiniteimally, we may wite thi a x x = g x, g G (11) x = x + δ x, (1) whee δ x = f ( x; ε) ae mooth function of the coodinate andε 1, = 1,,,dim(G), ae the infiniteimal goup paamete We aume that ou dynamical vaiable (the field) ae decibed by (1,3) n continuou function {1,,, n}, ϕ ( x) l ( M, C ) and tanfom by finite-dimenional ( n n matix) epeentation ρ of the goup (1,3) ( ρ( G), M ), (1,3) G on M : ϕ ( x) ϕ ( x) = D ( g) ϕ ( x), (13) D ( g g ) = D ( g ) D ( g ) (14) t 1 1 t Cloe to the identity element of the goup, ie infiniteimally, we can wite D ( g) = δ + ε ( Σ ), (15) whee the n n matice Σ fom a epeentation of the Lie algeba of the goup Uing (15), we can ewite (13) a [ Σ, ] ic γ Σ β = β Σ γ (16) ( x) = ( x) ( x), (17) δ ϕ ϕ ϕ whee ( x) ( Σ ) ( x) (18) δ ϕ ε ϕ 1

2 Fo late ue, let u now deive ome of the impotant popetie of the vaiation ymbol δ Fom (16) and (18), we ee that γ [ δ, δβ ] ϕ( x) icβ δγ ϕ( x) = (19) Hee, we defined the vaiation ymbol δ by ε δ ( ) = δ ( ) Uing it definition (17), we can eaily how that δ act a deivation on function δ ( f ( x) g( x)) = ( δ f ( x)) g( x) + f ( x)( δ g( x)) (110) Since δ ϕ i witten entiely in tem of the pimay field without time deivative, it (anti)commute with ϕ ( y) at equal time [ δ ϕ( t, x), ϕ ( t, y)] = 0 (111) To fit ode in δ x we can wite ϕ ( x) = ϕ ( x) + δ x ϕ ( x) (11) Uing thi expanion, we can ewite (17) a ( x) = ( x) + x ( x), (113) δ ϕ δϕ δ ϕ whee δϕ ( x) ϕ ( x) ϕ ( x), (114) i the infiniteimal change in the functional fom of field If the field index i a pace-time index taking value in the et {,, ν, νρ,}, in 4- dimenional pace-time δϕ ( x) i nothing but the Lie deivative of a pacetime teno ϕ ( x) with epect to the infiniteimal tanfomation (1) The tanfomation conideed i called pace-time (ymmety) tanfomation if δ x 0 and intenal (ymmety) tanfomation if x δ = 0 ; ( δ δ) i called the obital pat of the (ymmety) tanfomation Commuting both ide of (113) with ϕ ( t, y) (at equal time) and uing (111), we find 0 [ δϕ( t, x), ϕ( t, y)] = δ x [ ϕ( t, x), ϕ( t, y)] ɺ (115)

3 And finally, we note that it will be ueful to wite (113) a an opeato equation with δ = δ + δ x, (116) ( δ, δ ) ae given by thei uual definition on function (17) and (114) Notice that, while δ commute with deivative, δ doe not commute with : [, ] ( x δ δ ) = (117) Execie (11) In D-dimenional Minkowki pace-time, how that ( ) δ x = + ε x + ω x ν + c x ν x η ν x, i the mot geneal olution to the confomal Killing equation ν ν ν ( δ x ) + ( δ x ) = η ( δ x ) D Solution i given in [1] Execie (1) Let f ( x) be a olution to the confomal Killing equation ν f + f = ω( x) g ν ν ν Show that f x ɺ i a contant along the geodeic d dτ Solution: ( ) ( ) ( ) ν 1 f xɺ = xɺ f xɺ = f + f xɺ xɺ = ωg xɺ xɺ = 0 3

4 0 Invaiance of the Action and Noethe Theoem Emmy Noethe [ ] The action [ S :( ρ( G), Ω) R] i given by the integal S ϕ d xl ϕ x ϕ x 4 [ ] = ( ( ), ( )) Ω 4 (1,3) whee Ω R i an abitay, contactible and bounded egion in ( M, η) The Lagangian denity L ( x) i a local eal function of fundamental (pimay) field ϕ ( x), that i, it i contucted fom ϕ ( x) and thei deivative, ϕ, at the ame pace-time point x In ode fo the canonical fomalim to make ene, L ( x) i aumed to contain no econd o highe n 4n deivative of ϕ ( x) and to be at mot quadatic in ϕ ( L : C C R ) It i alo aumed that the field ae well-behaved and vanih ufficiently apidly at infinity Unde the tanfomation (11), a egion (1,3) D M i mapped into a new egion D by point-to-point coepondence Theefoe, infiniteimally g( ε ) induce an infiniteimal change in the egion of integation g( ε): Ω Ω = Ω+ δω 4

5 and map the action integal into g S S = d xl x (1) 4 ( ε): [ ϕ] [ ϕ] ( ) Ω+ δ Ω whee L( x) L ( ϕ ( x), ϕ ( x)) If the action integal i invaiant (ie, S[ ϕ ] = S[ ϕ ], ε andϕ ) unde cetain goup of tanfomation (with paamete ε ), the theoy i aid to have a ymmety coeponding to thoe tanfomation Theefoe, in ode fo the goup G to be a ymmety goup of ou theoy, we mut have δ ( ) d xl( x) = d xl( x) d xl ( x) = 0 () Ω Ω+ δ Ω Ω Changing the dummy integation vaiable in the econd integal, we get 4 4 d x x d x x Ω+ δ Ω Ω L( ) L ( ) = 0 (3) Uing the elation L( x) = L( x) +δl ( x), (4) we can ewite (3) in the fom ( δ ) d x x d x x d xδ x Ω+ Ω Ω Ω We know fom elementay calculu that L( ) L( ) + L ( ) = 0 (5) d dx f ( x) dx f ( x) f ( b) δb f ( a) δa = dx f ( x) δ x dx b+ δb b b a+ δ a a a ( ) (6) Genealizing thi to the 4-dimenional cae in (5), we find (to fit ode in δ x ) that 4 d x( ( Lδ x ) + δl ( x) ) = 0 (7) Ω The ame eult can be obtained if we apply the deivation popety of δ on the left-hand ide of () Uing the elation ( ) ( ) δ d xl( x) = δ d x L( x) + d xδl ( x) = 0 (8) Ω Ω Ω 5

6 det 1 + = exp( T In 1 + ) = exp( T ) 1+ T( ), we can expand the Jacobian of the tanfomation to fit ode a Thu x ρ ρ ρ τ det = det( δ + δ x ) 1+ T( δ x ) = 1+ τδ x x x = det = + ( δ ), δ ( ) = ( δ ) (9) x d x d x d x x d x d x d x x And applying the opeato equation (116) to the Lagangian, give u δ = δ + δ x L L L (10) Putting (9) and (10) in equation (8) give u back equation (7) 4 Since Ω R i an abitay contactible egion, the integand in equation (7) mut vanihe identically ( δ x ) 0 δl+ L (11) Thi i an identity with epect to all it agument, if the goup G i an invaiance goup of the action integal It hold fo all function ϕ ( x), it doe not matte whethe ϕ ( x) ae olution of the Eule-Lagange equation o not Since the divegence tem i not dicaded, the behaviou of ϕ ( x) on the bounday, Ω, i alo ielevant Since [ δ, ] = 0, we can wite L L δl ( x ) = δϕ + ( δϕ ) (1) ϕ ϕ Intoducing the Eule deivative ( ) ˆ δl L L δ S[ ϕ] ˆ =, (13) δϕ ϕ ϕ δϕ into equation (1) and ineting the eulting equation back into equation (11), we aive at the Noethe Identity ˆ δl L δϕ + x L δ + δϕ 0 (14) ˆ δϕ ( ϕ ) Let u aume the field ϕ ( x) ae olution of the Eule-Lagange equation 6

7 ˆ δl = 0, = 1,,, n (15) ˆ δϕ ( x) Thu, the Noethe identity implie that the object (Noethe cuent) L J ( x; ε) L δ x + δϕ, (16) ( ϕ ) atifie the conevation law = 0 (17) J The exitence of the coneved (ymmety) cuent J ( x) i known a the fit Noethe theoem Of coue, phyical cuent do not depend on the paamete of the ymmety goup Indeed, they can be factoed out of equation (16) leaving a goup index on the phyical cuent ( ) L J, 1,,,dim(G) ( ) x = δϕ + L δ x = (18) ( ϕ ) Thu, thee ae a many coneved cuent a thee ae paamete Notice that the coneved cuent i not unique In fact, adding a total divegence of antiymmetic teno to the Noethe cuent doe not poil the conevation law, J = J + F, F = F (19) ν ν ν ( ) ( ) ν ( ) ( ) ( ) ν Quantitie like thee F ae called upe-potential They play impotant ole in the contuction of the coneved chage in geneal elativity and othe geneally covaiant theoie Below, we will ue thi feedom to contuct the ymmetic enegy-momentum teno out of the non-ymmetic canonical teno It will be intuctive to give anothe deivation of the Noethe cuent which i ideologically diffeent fom the one we have jut done It conit in compaing the change δl without uing the equation of motion (ie the off-hell vaiation) with that when the Eule-Lagange equation ae atified (ie with the on-hell vaiation of L ) In thi method, the explicit fom of L a well a δϕ ae aumed known Accoding to (1), an abitay infiniteimal change in the field induce the following change in the Lagangian L L L δl = δϕ + δϕ ϕ ( ϕ) ( ϕ) Uing the equation of motion (ie on-hell), we find 7

8 L δl = δϕ (0) ( ϕ) Thi i alway tue, whethe o not δϕ i a ymmety tanfomation On the othe hand, if without the ue of equation of motion (ie jut by ubtituting δϕ in (1)) the Lagangian change by a total divegence of ome object Λ, L, (1) δ = Λ then the action emain unchanged, and the tanfomation, δϕ, i a ymmety tanfomation of the theoy Fom (19) and (0), it follow that the coneved Noethe cuent i L J ( x) = δϕ Λ ( x) ( ϕ ) () Notice that uing thi method to deive the cuent, you do not need to know how x tanfom unde the ymmety goup in quetion So, how can you ditinguih between intenal and pace-time ymmetie, if you don t have δ x? To anwe thi quetion we ue the fact that adding total divegence to a Lagangian doe not affect the dynamic Indeed, it i alway poible to find a dynamically equivalent Lagangian L ˆ( x) uch that ˆ( x) ( x ) L = L + (3) Thu ˆ δl = δl + δ = ( Λ + δ ) (4) So, if Λ = 0, o it i poible to emove it by a uitable choice of you ae dealing with an intenal ymmety Othewie, it i pace-time ymmety, then 1 Tanlation Invaiance and the Ste Enegy-Momentum Teno Uing the elation δ ϕ = δ ϕ δ ϕ, (5) ν ( x) ( x) x ν ( x) which follow fom (116) when we facto out the infiniteimal paamete ε, we can ewite (18) in the fom 8

9 whee ( ) L J ( ) ( ) x x x ν = δϕ θ ν δ, (6) ( ϕ ) L θ ν ( x) νϕ δ νl ( x) (7) ( ϕ ) Intoducing the canonical conjugate field L( x) π ( x) ɺ ϕ ( x), (8) 00 into the θ component, we find H ɺ L (9) 00 ( x) θ ( x) = π ( x) ϕ( x) ( x) In the claical mechanic of continuou ytem, H ( x) i nothing but the Hamiltonian denity of the ytem Theefoe, it follow fom conideation of ν covaiance that θ epeent the canonical te enegy-momentum teno of the field Let u calculate it divegence on hell, ie by uing the Eule- Lagange equation of motion, ν L ν L ν ν ν L θ = ϕ + ϕ L ( ϕ, ϕ, x) = η ϕ ( ϕ ) x Clealy, thi vanihe (on hell) if the Lagangian doe not depend explicitly on x Thu, the enegy-momentum teno i the coneved Noethe cuent aociated with the ymmety goup of pace-time tanlation Indeed, unde pace-time tanlation ν ν ν ν δ x = ε, δ x = δ (30) And all field, egadle of thei tenoial chaacte, ae invaiant, δ ϕ ( x) = 0 (31) Putting (4) and (5) into the defining equation (0) of J ( )( x), we find J ( x) θ ( x) Thu, it follow tivially fom (17) that the enegy-momentum teno i coneved ν θ ( x) = 0 (3) 9

10 Loentz Invaiance and the field Moment Teno Unde infiniteimal Loentz tanfomation δ x = η ω x, (33) ν ν ρ ρ the field tanfom accoding to i ν δ ϕ( x) = ω ( Σ ν ) ϕ( x), (34) whee Σ ν ae the appopiate (pin) tanfomation matice fo the field ϕ Uing thee elation, we can wite the coneved cuent, (6), in the fom 1 L J = ω ν i ( Σ ) ( x ρ x ρ ν ϕ ηρ θ ν ηνρ θ ) ( ϕ) Thi implie that the moment teno M L = ( η x θ η x θ ) i ( Σ ) ϕ ( ϕ ) ρ ρ ν νρ ρ ν ν = M ν (35) i coneved Thu, a field theoy i Poincae invaiant if and only if The object ν ν ν ν M = θ θ + S = 0 (36) ρν L ν S ( x) i ( Σ ) ϕ( x), (37) ( ϕ ) ρ depend entiely on the intinic popetie (tenoial natue) of the field In claical field theoy, it chaacteize the polaization popetie of the field Thu, it coepond to the pin (angula momentum) of paticle decibed by the quantized field Fo a ingle component field (cala), the (pin) matix Σ vanihe and (35) become M ν = x ν θ x θ ν Thi elation i vey muck like the elation between momentum and obital angula momentum in the claical mechanic of paticle xi p j x j pi = ijk Lk Theefoe, the fit tem in (35), L ν = x ν θ x θ ν, (38) hould coepond to ome intinic obital angula momentum of the field Thu the moment teno 10

11 νρ νρ νρ M = L +S (39) mut be elated the total angula momentum teno of the field Fom (35) and (36) we can clealy ee the connection between the ymmety popetie of the canonical enegy-momentum teno θ ν and the tuctue of the total moment teno M Fo a cala field, ince S 0, νρ νρ = we find that θν = θν On the othe hand, fo abitay multi-component field, θ ν i not neceaily ymmetic Howeve, the Poincae invaiance condition, (36), togethe with the feedom of adding upe-potential to the cuent, (19), allow u to contuct a ymmetic enegy-momentum teno ν ν Since S = S, let u eek fo a quantity F ν uch that ν ν ν S = F F (40) Putting thi in the Poincae invaiance condition (36), we find θ + F = θ + F ν ν ν ν Theefoe, if we et T ν ν ν θ + F, (41) we ee that T = T (4) ν ν Futhemoe the conevation of the canonical enegy-momentum teno, = 0, implie ν θ T ν ν = F (43) Thu, the conevation of the ymmetic teno T ν demand (upe-potential) ν ν F = F (44) Now, we can ue (40) and (44) to olve fo the upe-potential ν 1 ( ν ν ν F = S S + S ) (45) The coneved teno T ν i called the Belinfante ymmetic enegymomentum teno In tem of thi ymmetic teno, the total moment teno become 11

12 M = ( x T x T ) + ( F F S ) + P, ν ν ν ν ν ν ρν ρ P = F x F ρν ρ ν ρν x The quantity in the middle backet vanihe when (40) and (44) ae ued, and if we dop the upe-potential P, we aive at the Belinfante moment teno M ν = x ν T x T ν (46) Notice that the (intenal) pin pat ha diappeaed fom the total moment teno whoe conevation now follow fom the conevation of the ymmetical enegy-momentum teno The phyical impotance of the ymmetical (Belinfante) enegy momentum teno follow fom the belief that gaviton couple to T ν ν, and not to θ Indeed, if the matte theoy i minimally coupled to gavity and it action i vaied with epect to the metic teno g ν, the ymmetic enegy momentum teno i obtained in the limit g ν η ν δ D T = d x ( ϕ, ϕ, g ) ν ν g δ g L (47) ν With the ymmetical enegy momentum teno, the coneved Poincae cuent take on the compact (Beel-Hagen []) fom Since δ x J T x ν = νδ (48) = + ω x, we can ewite the cuent in the fom ν ν ν λ λ o 1 J ν T ν x T ν T ν = ν + ω ν = ν ω ( xν T x Tν ), (49) J 1 = ν T ω ν ν ν M (50) Taking the divegence of the cuent (48), and uing the ymmety of the enegy momentum teno, we find 1 J ( T ) x ν T ( x ν ν x = ν δ + ν δ + δ ) = 0 The fit tem on the ight hand ide vanihe becaue of T ν = 0, and the econd tem vanihe becaue Poincae tanfomation δ x f ( x) = + ω x ν, (51) ν 1

13 ae the mot geneal olution to the Killing equation (in any numbe of dimenion) Now conide the cuent f + f = 0 (5) ν ν J T x ν = νδ, (53) whee T ν i the coneved and ymmetic enegy momentum teno, and ν ν δ x = f i vecto field atifying the confomal Killing equation in D- dimenional pace-time Taking the divegence of the cuent, we find ν ν ν f + f = f η (54) D f J = T (55) D Thu, the cuent (53) i coneved povided that the coneved and ymmetic enegy momentum teno i tacele T = 0 Indeed, confomal invaiance allow u to futhe impove the Belinfante enegy-momentum teno o that it i tacele much in the ame way that Poincae invaiance allowed u to make the canonical Enegy-momentum teno ymmetic [1] Thi way, the coneved confomal cuent can be witten in the compact Beel-Hagen fom (53) 3 Poincae and Scale Invaiance and the Tacele Enegy Momentum Teno Unde cale (dilatation) tanfomation the field tanfom accoding to x x = e x, δ x = x, (56) and ϕ ϕ = ϕ δ ϕ = ϕ (57) ( ) ( ) d x x e ( x), ( x) d ( x) ( ) δϕ ( x) = d + x ϕ ( x) (58) whee the eal numbe d epeent the caling dimenion of the field ϕ (we aume all component have the ame caling dimenion) In D dimenion, it value fo cala and vecto field, i given by 13

14 D d = (59) Uing (6), we find the canonical dilatation cuent ν L D = θ ν x + dϕ ( ϕ ) (60) Thu L D = θ + dϕ (61) ( ϕ) Theefoe, in cale invaiant theoie, the tace of the canonical enegymomentum teno i given by L θ = dϕ (6) ( ϕ) We will now how that Poincae and cale invaiance allow u to contuct coneved, ymmetic and tacele enegy momentum teno Recall that fo ν ν any ank-3 teno R = R, anti-ymmetic in the lat two indice, the following combination i a teno anti-ymmetic in the fit two indice ν 1 ( ν ν ν ) ν F = R + R R = F (63) Thu, F ν = 0 and the following modified enegy-momentum teno i coneved Taking the tace, T T 1 = θ + ( R + R R ) (64) ν ν ν ν ν = η T, we find ν ν = θ + η R (65) T ν ν Uing the condition fo cale invaiance, (6), the tace become T L = η R dϕ (66) ( ϕ) ν ν Recall that, in ode to ymmetize the enegy-momentum teno, Poincae ν ν invaiance allowed u to chooe R = S, ee (45) So, fo Poincae and 14

15 cale invaiant theoie, tacele enegy-momentum teno can be obtained by witing ν ν 1 ν ν R = S + ( η V η V ), (67) D 1 whee the vecto field V i detemined fom (66) by etting T = 0: L V = dϕ η ( ρϕ) νρ ρ νs (68) Notice that fo a cala field, the pin matix vanihe Thu fom (37) we find S ν = 0 Theefoe, if we conide the cale invaiant field theoy we find D 1 ν D L = ην ϕ ϕ λϕ, (69) D D ϕ ϕ 4 4 V =, V = (70) Thu, the coneved, ymmetic and tacele enegy-momentum teno i D 4( D 1) ( η ) ν ν ν ν T = T + ϕ, (71) whee T ν i the ymmetic enegy-momentum teno T ν = ϕ ν ϕ L (7) Execie (1) Show that the enegy momentum teno (71), i coneved, ν ν T = 0 and tacele, η T = 0 Execie () Conide the fee Maxwell theoy in D-dimenional pace-time ν 1 L = 4 F F ν ν Show that the ymmetic, coneved enegy momentum teno i given by the Belifante expeion in any numbe of dimenion but tacele only in 4 dimenion 1 T F F F 4 ν ν ν = + η, 15

16 T D = 1 F 4 Execie (3) Show that Maxwell theoy i cale invaiant in any numbe of dimenion Given that D δ A = x + A Show that (up to a tivially coneved upe potential) the coneved dilation cuent i given by ν 4 D ν D = T ν x + F Aν By explicit calculation, how that D = 0 4 The New Impoved CCJ teno [3] and the Belinfante teno fom the action: fun with upe-potential Baically, thi ubection i about obtaining the eult of the lat two ubection fom vaying the action integal Recall that on-hell, the vaiation of the action can be witten a L δ = δ ϕ θ δ 4 ν S d x x ν ( ϕ) (73) To thi vaiation, we will add zeo and how that the coefficient of δ x ν i the new impoved enegy-momentum teno of Callan, Coleman and Jackiw [3] Conide the functional ( ), 4 R = d x Γ x whee Γ i a local function of the field and it deivative to finite ode We vay thi uing (110), 4 4 δ R = δ d x Γ+ d xδ Γ ( ) ( ) If in the fit integal we make ue of (9), and in the econd integal we ue (116), we find 16

17 ( )( ) ( ) ( )( ) R = d x x Γ + d x Γ + d x x Γ δ δ δ δ Combining the fit and the thid tem and uing [ δ, ] = 0 in the econd tem, we get ρ ( ρ ) R = d x Γ + x Γ (74) 4 δ δ δ Uing (116) again, thi can be ewitten a ρ ρ τ ( ( ) ) 4 δ R = d x δ Γ + δτ ργ δτ ργ δ x (75) Integating the econd tem by pat, we find ρ ρ τ ρ ρ { ( ) } ( ) δ R = d x δ Γ δ Γ δ Γ δ x + δ x Γ δ x Γ 4 τ τ ρ ρ The econd tem vanihe identically, and we ae left with ρ ρ τ { ( ) } 4 δ R = d x δ Γ δτ Γ δτ Γ ρδ x (76) Now, we add (76) and ubtact (75) fom (73) 4 L ρ ρ τ δ S = d x δ ϕ ( δτ Γ δτ Γ ) ρδ x ( ϕ) ρ ρ τ {( θ δ δ ) δ x } 4 d x τ + τ ργ τ ργ (77) Let u aume that ou ymmety tanfomation change the action accoding to Now, if we can find Γ uch that S d x ( x) (78) 4 δ = Λ L ( ϕ ) ( ) Γ Γ x = Λ (, ) ρ ρ τ δ ϕ δτ δτ ρδ ϕ ϕ, (79) hold, then the Noethe cuent will have the Beel-Hagen fom τ J = T τδ x, (80) with a new impoved enegy-momentum teno given by T ( ) = θ + η Γ η Γ (81) τ τ τ ρ ρτ ρ 17

18 Execie (4) Conide the fee male cala field action in fou dimenion S 1 ϕ ϕ 4 = d x (i) Show that δ S = 0, ie Λ ( x) = 0, unde the Loentz tanfomation δ x = ω δ ϕ = ν ν x, 0 Then how that Loentz invaiance implie the following condition on ν ν Γ = Γ (ii) Show that δ S = 0, Λ = 0, unde the cale tanfomation Γ x = x, = δ δ ϕ ϕ Ue (79) to how that cale invaiance implie 1 6 Γ = ϕ Ue thi to how that (81) lead to the ymmetic and tacele enegymomentum teno T 1 = θ + ( η ) ϕ 6 ν ν ν ν (iii) Show that the confomal tanfomation δ x = x c x c x δ ϕ = c x ϕ, ( ), ( ) change the cala field action by total divegence ( ϕ ) δ S = d x Λ = d x c 4 4 Ue Loentz invaiance and cale invaiance etiction on (79) i atified fo the confomal tanfomation Γ to how that 18

19 30 Canonical Quantization and Equal-Time Commutato Algeba Befoe intoducing the Noethe chage and dicu it popetie, we need to et up ou quantization cheme We will wok with the canonical quantization ule: Poion backet of function goe to equal-time (anti)commutato of the coeponding opeato 1 i{ f ( t, x), g( t, y)} [ fˆ ( t, x), gˆ ( t, y )] PB Howeve, we will not ue the hat to indicate opeato Alo, we will not wite the time agument explicitly Unle tated othewie, all (anti)commutato ae conideed to be at equal time So, fo any local functional F, we ue the following ule δ F( x) i{ F( x), ϕ( y)} PB = i = [ F( x), ϕ( y)], δπ ( y) (31) δ F( x) i{ F( x), π ( y)} PB = i = [ F( x), π ( y)] δϕ ( y) (3) The fundamental equal-time commutation elation follow by putting F = ϕ, π : ϕ π δ δ 3 [ ( x), ( y)] = i ( ), [ ϕ ( x), ϕ ( y)] = [ π ( x), π ( y)] = 0 x y (33) Notice that π ( x) = ( ϕ, ϕ, ɺ ϕ) ɺ ϕ L (34) In mot cae, the Lagangian i quadatic inϕɺ Thu, π i linea inϕɺ We will aume that the imultaneou linea equation (34) can be olved foϕɺ That i we aume that it i poible to epeent the velocity by ome function ɺ ϕ ( x) = f ( ϕ, ϕ, π) (35) The functional deivative of any local function of the field vaiable, uch a the Lagangian, will be calculated uing the chain ule: 1 In ef [4] I have out-lined the poof of lim i ( ˆ ˆ ˆ ˆ ) A AB BA B A = B ħ 0ħ x p p x 19

20 ( x) ( j ϕ) δl( x) L( x) δϕ( x) L( x) δ L( x) δϕɺ ( x) = + +, δϕ ( y) ϕ ( x) δϕ ( y) ( ϕ ) δϕ ( y) ɺ ϕ ( x) δϕ ( y) j L( x) L( x) δϕɺ ( x) = δ ( x y) + δ ( x y) + π ( x), ϕ ( ) ( ϕ ) δϕ ( y) 3 ( x) 3 j x j δl( x) L( x) δϕɺ ( x) δϕɺ ( x) = = π ( x) δπ ( y) ɺ ϕ ( x) δπ ( y) δπ ( y) Thu, uing (31) and (3) in the above two equation, we infe the following equal-time commutation elation [ L ( x), ϕ ( y)] = π ( x)[ ɺ ϕ ( x), ϕ ( y)], (36) π δ x y L δ x y L L π ɺ ϕ π (37) 3 ( x) 3 [ ( x), ( y)] = i ( ) + i j ( ) + ( x)[ ( x), ( y)] ϕ ( jϕ) The lat two tem in (37) can be combined to give L( x) L( x) L (38) 3 [ ( x), π ( y)] = i δ ( x y) + [ ϕ( x), π ( y)] ϕ ( x) ( ϕ) We will now ue thee commutation elation togethe with the fundamental equal-time commutation elation (33) to deive few impotant equation Let tat by deiving the o-called Heienbeg equation [ ih, ϕ ( x)] = ɺ ϕ ( x), [ ih, π ( x)] = ɺ π ( x) (39) The HamiltonianH i obtained by integating the Hamiltonian denity which we defined in (9) = d xh x = d xθ x, (310) H ( ) ( ) wheea the 3-momentum of the field i given by = d xθ x (311) j 3 0 j P ( ) So, let u commute the Hamiltonian denity with ϕ ( y) and π ( y) [ H( x), ϕ ( y)] = [ π ( x), ϕ ( y)] ɺ ϕ ( x) + π ( x)[ ɺ ϕ ( x), ϕ ( y)] [ L( x), ϕ ( y)], [ H( x), π ( y)] = π ( x)[ ɺ ϕ ( x), π ( y)] [ L( x), π ( y)] Ineting (36) and (37) and uing (33), we find 0

21 ih x ϕ y = δ x y ɺ ϕ x, (31) 3 [ ( ), ( )] ( ) ( ) L( x) L( x) H π δ x y δ x y (313) 3 ( x) 3 [ i ( x), ( y)] = ( ) + j ( ) ϕ ( jϕ) Integating ove x, we find (afte integating (313) by pat) [ ih, ϕ ( y)] = ɺ ϕ ( y), L( y) L( y) L ( y) [ ih, π ( y)] = j = 0 = ɺ π ( y), ϕ( y) ( jϕ) ɺ ϕ whee the Eule-Lagange equation and the definition of the conjugate momentum (8) have been ued In QFT thee ae called Heienbeg equation which, accoding to (31) and (3), coepond to the canonical Hamilton equation in claical field theoy δh δh = ɺ ϕ( x), = ɺ π( x) (314) δπ ( x) δϕ ( x) In exactly the ame way, we can etablih the following moe geneal equaltime commutation elation iθ x ϕ y = δ x y ϕ x (315) 0 3 [ ( ), ( )] ( ) ( ), iθ x π y = η L( x) η π x δ x y + η L( x) δ x y (316) 0 0 j [ ( ), ( )] ( ) j ( ) ( ) jϕ ϕ Integating thee ove x and intoducing the Eule deivative (13) in (316), we find whee i t ϕ x i d x θ x ϕ x = ϕ x, (317) 3 0 [ P ( ), ( )] [ ( ), ( )] ( ) ˆ [ ip ( t), π ( x)] [ i d x θ ( x ), π ( x)] π ( x) η δ L = +, (318) ˆ δϕ t = d xθ x, (319) 3 0 P ( ) ( ) i the field enegy-momentum 4-vecto We will talk about the phyical meaning of thee elation, and how thatp i indeed a time-independent 4- vecto, when we intoduce the Noethe chage We will alo ee why the Eule-Lagange equation of motion appea in (318) but not in (317) Notice that (318) how the equivalence between Lagangian and Hamiltonian fomalim 1

22 ˆ δl( x) δp = 0 [ ip, π ( x)] = π ( x) = ˆ δϕ ( x) δϕ ( x) (30) Okay, ince we calculated the ETCR of the 0-th component of tanlation 0 cuent, θ, with the field, let u do ome jutice to the Poincae goup and oν do the ame thing with the 0-th component of Loentz cuent M Fom (35) thi i given by M ν ( x) = x ν θ ( x) x θ ν ( x) iπ ( x)( Σ ν ) ϕ( x) (31) Uing (315) and the fundamental ETCR (33), we find ( ) im x ϕ y = x ϕ x ϕ i Σ ϕ x δ x y (3) 0ν ν ν ν 3 [ ( ), l( )] l l ( ) l ( ) ( ) Integating ove x and changing y x, we get ou final eult i t ϕ x i d x M x ϕ x = x ϕ x ϕ i Σ ϕ ν 3 0ν ν ν ν [ M ( ), ( )] [ ( ), ( )] ( ) (33) In the next ection we will how that the integal t = t + t = d xm x, (34) ν ν ν 3 0ν M ( ) L ( ) S ( ) ( ) i a time-independent Loentz teno It epeent the total angula momentum of the field, with the obital angula momentum L ν and the pin angula momentum S ν ae given by ( θ θ ) ν 3 oν 3 0ν ν 0 L ( t) d xl ( x) = d x x ( x) x ( x), ν 3 0ν 3 ν S ( t) d xs ( x) = i d xπ ( x)( Σ ) ϕ( x) (35) Befoe going any futhe, let u make the following obevation: If we contact (317) with the tanlation paamete a and (33) with the Loentz ν paameteω, we ecove the coect infiniteimal Poincae tanfomation on the field δϕ ( x) = [ ia P ( t), ϕ ( x)] = a ϕ ( x), (36) i ν ν i δϕ( x) = [ ω M ν ( t), ϕ( x)] = ω ν x ϕ ( ω Σν ) ϕ( x) i = δ x ϕ ( ω Σ) ϕ( x) Remakably, the only aumption ued to deive thee elation i (that the field atify) the fundamental ETCR (33) Indeed, neithe dynamical conideation (ie, equation of motion) no ymmety conideation (ie, (37)

23 conevation law) have been ued to obtain (36) and (37) In fact we will now how that the ame eult hold fo abitay goup of tanfomation Setting = 0 in the Noethe cuent (18), we find J( )( x) = π ( x) δ ϕ ( x) δ x L ( x) (38) 0 0 Thu, at x = y 0 0 [ J ( x), ϕ ( y)] = π ( x)[ δ ϕ ( x), ϕ ( y)] + [ π ( x), ϕ ( y)] δ ϕ ( x) δ x [ L ( x), ϕ ( y)] 0 0 ( ) Due to (115) and (36) the fit and the thid tem add up to zeo, and the econd tem, when evaluated fom the ETCR (33), give [ ij ( x), ϕ ( y)] = δ ϕ ( x) δ ( x y ) (39) 0 3 ( ) Hence, a 3-volume integation ove x will geneate the infiniteimal tanfomation on the field δ ϕ ( x) = [ iq ( t), ϕ ( x)], (330) whee Q ( t) (the Noethe chage) i given by the integal Q ( t) = d x J ( x) (331) 3 0 ( ) Again, the impotant fact about (330) i that it ha been deived fo abitay goup of tanfomation and without efeence to a pecific fom of L ( x), ie, without any commitment to ymmety o dynamic Theefoe, even when the tanfomation i not a ymmety opeation and at leat fomally, it eem that (330) and it local veion (39) ae alway valid Thu, any eult that can be deived fom them will neceaily be tue if we ignoe the uual difficultie of local quantum field theoy The o-called Wad-Takahahi identity in QFT i jut an altenative fom of the canonical equal-time elation (330) It can be deived fom the following timeodeed poduct ( ) T J ( x) ϕ ( y) = Θ( x y ) J ( x) ϕ ( y) +Θ( y x ) ϕ ( y) J ( x) (33) ( ) ( ) ( ) Diffeentiating thi with epect to x and uing the elation Θ( x y ) = η δ( x y ), ( x) Θ( y x ) = η δ( x y ), ( x) (333) 3

24 we find ( ) ( ) T J ( x) ϕ ( y) = T J ( x) ϕ ( y) + δ ( x y )[ J ( x), ϕ ( y)] (334) ( ) ( ) ( ) The peence of the delta function allow u to ue (39) in the econd tem on the ight and get 4 ( ) ( ) T J ( x) ϕ ( y) = T J ϕ ( y) iδ ( x y) δ ϕ ( x) (335) ( ) ( ) ( ) Now, on integating (335) ove a pace-time egion Ω containing the point y, we aive at the Wad-Takahahi identity 4 4 ( ( ) ) ( ( ) ) Ω (336) Ω iδ ϕ ( y) = d xt J ( x) ϕ ( y) d x T J ( x) ϕ ( y) Since the vaiation δ doe not upet the time odeing, n ( 1 n ) ( 1 i n ) iδt ϕ ( y ) ϕ ( y ) ϕ ( y ) = i T ϕ ( y ) δ ϕ ( y ) ϕ ( y ), (337) 1 n 1 i n i= 1 we can genealize (336) and obtain, fo all yi Ω, a geneal fom fo the W-T identity 4 4 T ( ϕ ( ) ( ) ( ) 1 1) δ ϕ ( ) T ( ) ( 1 1) T ( ) ( 1 1) i i = ϕ ϕ i y y d x J x y d x J x y i= 1 Ω (338) It i clea that the fit tem on the ight hand ide of (336) and (338) vanihe if the cuent i coneved, ie, when the tanfomation i a ymmety tanfomation Let u now go back to (35) and ue it to build up the equal time commutato ν [ im, π ( x)] Fo the pin angula momentum, we find Ω Uing (33), thi become π π ϕ π ν 3 l ν [ is, ( x)] = d x ( x )( Σ ) l [ ( x ), ( x)] ν [ S, ( )] ν i π x = iπ ( x)( Σ ) (339) To calculate the commutato with the obital angula momentum, we multiply (316) by (ix ) and integate ove x Integation by pat then give 3 0ν ν j 0ν L( y) 0ν L( y) [ i d x x θ ( x), π ( y)] = η j ( y π ( y) ) j y η + y η ( jϕ) ϕ 4

25 (340) Uing the identitie η = η ν ν 0 ν j 0 η δ η = δ η ν 0ν jν 0 j j,, (341) in (338), we find afte ome eaangement 3 0ν ν ν 0ν L [ i d xx θ, π ( y)] = η π + y π η δ0 π + δ j ( jϕ) L y η ɺ π L ( ) ( ) 0ν + j ϕ jϕ In thi equation, we ue the definition of the conjugate momentum to obtain L( y) L( y) π ( y) =, ɺ π ( y) = 0 ( 0ϕ ) ( 0ϕ ), (34) L ˆ δl [ i d x x θ, π ( y)] = η π + y π δ η + y η (343) ˆ δϕ 3 0ν ν ν ν 0 ν 0 ( ϕ) Thu, the equied commutato follow by the following anti-ymmetic combination ( ) δl( x) [ il, ( x)] x x ˆ δϕ ˆ ν [ ν ] [ ν ]0 L x [ ν ]0 π = π δ η + η ( ϕ) (344) Adding (34) to (337), we get ν [ M, ν ( )] ν ( ) ( ) ν i x x x x x i ( x)( ) ν π = π π + π Σ + F (345) The exta tem in the Loentz tanfomation law fo π i given by F ˆ δl ( x F ) L ( x = x η δ η ) ˆ δϕ ( ϕ ) ν ν [ ν ]0 [ ν ]0 (346) Thi exta piece doe not vanih even when the equation of motion ae atified Thi i becaue the et{ π ( x)} doe not fom a covaiant manifold unde the Poincae goup 5

26 Now, let u ue Heienbeg equation (39) to pove the following theoem Theoem 31 If the equal-time commutation elation (33) ae valid at a 0 0 cetain time x = y = t, they ae alo valid at t + ε Poof: The poof will be baed on Heienbeg equation and the following Jacobi identity [ ϕ ( x, t),[ ih, π ( y, t)]] + [ π ( y, t),[ ϕ ( x, t), ih]] + [ ih,[ π ( y, t), ϕ ( x, t)]] = Since at x = y = t, we have 3 [ ϕ(, t), π (, t)] = iδ δ ( ) x y x y Thu the thid tem in the Jacobi identity vanihe becaue the delta ae c- numbe And we ae left with [ ϕ ( x, t),[ ih, π ( y, t)]] + [[ ih, ϕ ( x, t)], π ( y, t)] = 0 (347) Uing Heienbeg equation (39) in (345), we find [ ϕ ( x, t), ɺ π ( y, t)] + [ ɺ ϕ ( x, t), π ( y, t)] = 0 (348) Multiplying thi byε and adding the eult to the fundamental equal-time commutation elation, we find ɺ 3 [ ϕ (, ), (, )] [ (, ), (, )] [ (, ), (, )] x t π y t + ϕ x t επ y t + εϕ x t π y t = iδ δ ( x y) (349) Thu, to fit ode inε we can wite thi a ɺ o 3 ( ϕ t εϕ t ) ( π t επ t ) iδ δ ( x, ) + ɺ ( x, ), ( y, ) + ɺ ( y, ) = ( x y), (350) x y x y (351) 3 [ ϕ(, t + ε), π (, t + ε)] = iδ δ ( ) And, by exactly the ame method, we can etablih the emaining commutation elation [ ϕ ( x, t + ε), ϕ ( y, t + ε)] = [ π ( x, t + ε), π ( y, t + ε)] = 0 (35) 6

27 qed Refeence and Futhe Reading [1] Samalkhaiat Confomal Goup, Algeba and Confomal Invaiance in Field Theoy, Phyicfoumcom, thead numbe [] E Beel-Hagen, Math Ann 84, 58 (191) [3] C G Callan, J, S Coleman and R Jackiw, A New Impoved Enegy Momentum Teno, Ann Phy 59, 4 (1970) [4] Samalkhaiat Phyicfoumcom, thead numbe 7