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Scale Invariance, Renormalisation 158 9Renormalization of renormalizable theories without mass terms. These theories are scale invariant at the classical level because the action does not contain any dimensionful parameter. In this case the running of the coupling constants can be seen as resulting from a quantum breaking of classical scale invariance: different energy scales in the theory 158 are distinguished by different values of the9Renormalization coupling constants. Remembering what we learned in Section 8, we conclude that classical scale invariance is an anomalous of renormalizable theoriessymmetry. without massOne heuristic terms. These way theor to seeies how are scale the conformal invariant at anomaly comes about is to the classical level becausenotice the action that the does regularization not contain any of an dimensionful otherwise parameter.scale invariant field theory requires In this case the running158 the of introduction the coupling of constants an energy can scale be seen (e.g.as a resulting cutoff). This from9Renormalization bre a aking of scale invariance quantum breaking9.3 The ofRenormalization classical scale Group invariance: in Statistical different Mechanicsenergy scales in the theory 163 of renormalizablecannot be restored theories without after renormalization. mass terms. These theories are scale invariant at are distinguished by differentNevertheless, values of the scale coupling invariance constants. is not Remembering lost forever inwhat the quantum theory. It is re- we learnedmanaged in Sectionthe toclassical 8, rewrite we conclude level the partition because that classical the function action scale solely does inv not inariance termscontain is of any anthis anomalous dimensionful new (renormalized) parameter. In thiscovered case1 the at running the fixed of the points coupling of the constants beta function can be1 seen where,as resulting by defin fromition, a the coupling symmetry.spin One variables heuristic ways interacting to see how through the conformal a new Hamiltonian anomaly comesH about is to quantumdoes breaking not run. of classical To understand scale invariance: how this different happensenergy we go scales back in to the a sca theoryle invariant clas- notice that the regularization of an otherwise scale invari1 ant1 field theory requires are distinguishedsical field by theory different whose values field of!!H thex couplingstransforma const underants. coordinate Remembering rescalings what as 9.3 The Renormalization Group in Statistical Mechanicsthe introduction 161 of an energy scale (e.g.Z a cutoff). Thise breaking. of scale invariance (9.43) we learned in Section 8, we conclude that classical scale invariance is an anomalous square lattice as the one depicted in Fig 9.2. In terms of the spin variables s 1 , 1 cannot be restoredi 2 after renormalization. µ s µ # 1 where i labels the lattice site, the Hamiltonian of the system is given by symmetry. One heuristic wayx to see" howx , the conformal! x anomaly" !comes" x about, is to (9.23) Nevertheless, scale invariance is not lostScale forever invariance, in the quantum theory. renormalisation It is re- H J si s j, (9.35) notice that the regularization of an otherwise scale invariant field theory requires i, j covered at theLet fixed us pointsnow think of the about beta the function space of where, all possible by defin Hamiltoniansition, the couplingfor our statistical where i, j indicates that the sum extends over nearest neighbors and J is the cou- the introductionwhere # ofis ancalled energy the scale canonical (e.g. a scaling cutoff). This dimension breaking of of the scale field. invariance An example of such pling constant between neighboring spins (here we consider that there issystem no external including allRenormalisation kinds of possible deals couplings with the between scale thdependenceeindividualspinscom- of the the does not run. To understand how this happens we4 go back to a scale invariant clas- magnetic field). The starting point to study the statistical mechanics of this system cannotatheoryisamassless be restored after renormalization.! theory in four dimensions is the partition function defined as sical field theorypatible whose with the field symmetriesphysics! x transform even of the if the under system. original coordinate If denote theory rescalings by isR scalethe decimationinvariant. as operation, ! H Nevertheless, scale invariance is not lost forever in the quantum theory. It is re- Z e , our(9.36) previous analysis shows that R defines a map in this space of Hamiltonians si coveredµ at theµ fixed points of the beta# function11 where,µ byg defin4 ition, the coupling where the sum is over all possible configurations of the spins and 1 is the Virtual phenomena can get more complicated or simplify as ! T x "x , ! x " L! " x ,µ! ! !(9.23), (9.24) inverse temperature. For J 0theIsingmodelpresentsspontaneousmagnetization does not run. To understand how this happens1 2 we go back4! to a scale invariant clas- below a critical temperature Tc,inanydimensionhigherthanone.Awayfromthis we move to largerR : Hand shorterH . distances (9.44) temperature correlations between spins decay exponentially at large distances sical field theory whose field ! x transform under coordinate rescalings as wherexij # is called the canonical scaling dimension of the field. An example of such s s e " , (9.37) where the scalar field has canonical scaling dimension # 1. The Lagrangian den- i j At the same4 time the operation replaces a lattice with spacing a by another one atheoryisamassless! theory in fourµ dimensionsRµ # 1 with xij the distance between the spins located1 in the i-th and j-th sites of the lat- sity transformsx as "x , ! x " ! " x , (9.23) tice. This expression serves as a definitionλ of− the correlation length " whichwith sets the double spacing 2a.Asaconsequencethecorrelationlengthinthenewlattice characteristic length scale at which spins can influence eachotherbytheirinterac- " tion through their nearest neighbors.× measured in units of the1 latticeµ spacingg is4 divided by two,4R : " . where # is calledL theµ canonical! ! scaling! , dimensionL " of theL field.! (9.24) An2 example of such (9.25) Nowatheoryisamassless we can iterate2 the! operation4 theory4! inR fouran indefinite dimensions number of times. Eventually we might reach a Hamiltonian H that is not further modified by the operation R where the scalar field hasand canonical the classical scaling action dimension remains# invariant1. The3 Lagrangian. den- 1 µ g 4 sity transforms as We look at the freeL theoryµ! g ! 0foramoment.Nowtherearenodivergences! , (9.24) H R H 1 R H2 2 R ... 4!R H . (9.45) and all correlation functions can be exactly computed. In particular we consider the where the scalarL field has" canonical4L ! scaling dimension # 1. The(9.25) Lagrangian den- The fixed pointmomentum Hamiltonian spaceHn-pointis scale correlation invariant functionbecause it does not change as sity transformsIn asrelativistic QFT we seem to get only fixed points, no limit and the classicalR is performed. action remains Noticecycles invariant that nor because strange3. 4 of 4attractors this invariance the correlation length of the 9.4Fig. 9.3TheDecimation Renormalization of the spin lattice. Group Each in block Quantum in the upper Field lattice Theory is replaced by an effective G0 165p1,...,pn 2$ % p1 ... pn spin computed according to the rule (9.39). Notice also that the size of the lattice spacingsystem is doubled at the fixed point do not change under 4R.Thisfactiscompatiblewiththe in the process. We look at the free theory g 0foramoment.NowtherearenodivergencesL " L ! (9.25) ! 2 " and all correlationtransformation functions" can2 beonly exactly if " computed.0or" In pa.Herewewillfocusinthecaseofrticular we consider the 4 4 ip1 x31 ... ipn xn momentumnontrivial spaceandn-point the fixed classical correlation points action with functiond infinite remainsx1 ... correlationd invariantxn e . length. 0 T !0 x1 ...!0 xn 0 , (9.26) The spaceWe look of Hamiltonians at the free theory can beg parametrized0foramoment.Nowtherearenodivergences by specifying the values of the and4 all correlation4 functions can be exactly computed. In particular we consider the G0 p1,...,couplingpn 2$ constants% wherep1 associated by...!0 xpn we with denote all possible the free interacti field operator.on terms between Applying individual the rescaling (9.23) we F momentum space n-point correlation function spins of the lattice.find the If following we denote transformation by Oa si these (possibly for the correlation infinite) interaction function terms, 4 4 ip1 x1 ... ipn xn Luis Alvarez-Gaume Natal 23-26 Lectures March 2011 Luis the most general Hamiltonian for the spin system under study can be written as d x1 ...! d xn e 4 4 0 T !0 x1 ...!0 xn 0 , (9.26) O G01 p1,...,pn 2$ % p1 ... pn 4 n 1 n# Fig. 9.4 Example of a renormalization group flow. G0 p1,...,pn " G0 " p1,...," pn . (9.27) 4 4 ip1 x1 ... ipn xn where by !0 x we denote thed freex1 ... fieldd x operator.Hn esi Applying#aOa0s theTi , ! rescaling0 x1 ...!0 (9.xn23)0 we, (9.46)(9.26) perturbations the system flows either to the free theory at theoriginortoatheory 65 with infinite values for the couplings.find the following transformation for the correlationa 1 function Sunday, 27 March, 2011 wherewhere# R byare!0 thex couplingwe denote4 n constants1 then# free fieldfor the operator. corresponding Applying operators the rescaling.Thesecon- (9.23) we G0 ap1,...,3 Inp anD-dimensional" theoryG0 the" canonicalp1,...," scalingpn . dimensions(9.27) of thefieldscoincidewithitsengi- 9.4 The Renormalization Group in Quantum Fieldfind Theory the following transformation for the correlation function stants can beneering thought dimension: of as coordinates# D 2 for in bosonic the space fields of and all# HamDiltonians.1 for fermionic Therefore ones. For a Lagrangian the operation R defines a transformation2 in the set of coupling constants2 Let us see now how these ideas of the renormalization group apply to Fieldwith Theory. no dimensionful,..., parameters4 classicaln 1 n# scale invarianc,...,efollowsthenfromdimensionalanalysis.. Let us begin with a quantum field theory defined by the Lagrangian G0 p1 pn " G0 " p1 " pn (9.27) 1 R : #a # .
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