Introduction To Quantum Field Theory

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Introduction to Quantum Field TheoryJohn CardyMichaelmas Term 2010 – Version 13/9/10AbstractThese notes are intended to supplement the lecture course ‘Introduction to Quantum Field Theory’ and are not intended for wider distribution. Any errors or obviousomissions should be communicated to me at A Brief History of Quantum Field Theory22 The Feynman path integral in particle quantum mechanics42.1Imaginary time path integrals and statistical mechanics . .3 Path integrals in field theory793.1Field theory action functionals . . . . . . . . . . . . . . . .103.2The generating functional . . . . . . . . . . . . . . . . . .113.3The propagator in free field theory . . . . . . . . . . . . .144 Interacting field theories184.1Feynman diagrams . . . . . . . . . . . . . . . . . . . . . .184.2Evaluation of Feynman diagrams . . . . . . . . . . . . . .265 Renormalisation295.1Analysis of divergences . . . . . . . . . . . . . . . . . . . .295.2Mass, field, and coupling constant renormalisation . . . . .321

QFT126 Renormalisation Group6.1Callan-Symanzik equation . . . . . . . . . . . . . . . . . .406.2Renormalisation group flows . . . . . . . . . . . . . . . . .416.3One-loop computation in λφ4 theory . . . . . . . . . . . .446.4Application to critical behaviour in statistical mechanics .466.5Large N . . . . . . . . . . . . . . . . . . . . . . . . . . . .507 From Feynman diagrams to Cross-sections7.1The S-matrix: analyticity and unitarity . . . . . . . . . . .8 Path integrals for fermions139535862A Brief History of Quantum Field TheoryQuantum field theory (QFT) is a subject which has evolved considerablyover the years and continues to do so. From its beginnings in elementaryparticle physics it has found applications in many other branches of science,in particular condensed matter physics but also as far afield as biologyand economics. In this course we shall be adopting an approach (thepath integral) which was not the original one, but became popular, evenessential, with new advances in the 1970s. However, to set this in itscontext, it is useful to have some historical perspective on the developmentof the subject (dates are only rough). 19th C. Maxwell’s equations – a classical field theory for electromagnetism. 1900: Planck hypothesises the photon as the quantum of radiation. 1920s/30s: development of particle quantum mechanics: the samerules when applied to the Maxwell field predict photons. Howeverrelativistic particle quantum mechanics has problems (negative energystates.) 1930s/40s: realisation that relativity quantum mechanics, in whichparticles can be created and destroyed, needs a many-particle descrip-

QFT13tion where the particles are the quanta of a quantised classical fieldtheory, in analogy with photons. 1940s: formulation of the calculation rules for quantum electrodynamics (QED) – Feynman diagrams; the formulation of the path integralapproach. 1950s: the understanding of how to deal with the divergences of Feynman diagrams through renormalisation; QFT methods begin to beapplied to other many-body systems eg in condensed matter. 1960s: QFT languishes – how can it apply to weak strong interactions? 1970s: renormalisation of non-Abelian gauge theories, the renormalisation group (RG) and asymptotic freedom; the formulation of theStandard Model 1970s: further development of path integral RG methods: applications to critical behaviour. 1970s: non-perturbative methods, lattice gauge theory. 1980s: string theory quantum gravity, conformal field theory (CFT);the realisation that all quantum field theories are only effective oversome range of length and energy scales, and those used in particlephysics are no more fundamental than in condensed matter. 1990s/2000s: holography and strong coupling results for gauge fieldtheories; many applications of CFT in condensed matter physics.Where does this course fit in?In 16 lectures, we cannot go very far, or treat the subject in much depth.In addition this course is aimed at a wide range of students, from experimental particle physicists, through high energy theorists, to condensedmatter physicists (with maybe a few theoretical chemists, quantum computing types and mathematicians thrown in). Therefore all I can hope todo is to give you some of the basic ideas, illustrated in their most simplecontexts. The hope is to take you all from the Feynman path integral,through a solid grounding in Feynman diagrams, to renormalisation and

QFT14the RG. From there hopefully you will have enough background to understand Feynman diagrams and their uses in particle physics, and havethe basis for understanding gauge theories as well as applications of fieldtheory and RG methods in condensed matter physics.2The Feynman path integral in particle quantummechanicsIn this lecture we will recall the Feynman path integral for a system witha single degree of freedom, in preparation for the field theory case of manydegrees of freedom.Consider a non-relativistic particle of unit mass moving in one dimension.The coordinate operator is q̂, and the momentum operator is p̂. (I’ll becareful to distinguish operators and c-numbers.) Of course [q̂, p̂] ih̄.We denote the eigenstates of q̂ by q 0 i, thus q̂ q 0 i q 0 q 0 i, and hq 0 q 00 i δ(q 0 q 00 ).Suppose the hamiltonian has the form Ĥ 21 p̂2 V (q̂) (we can considermore general forms – see later.) The classical action corresponding to thisisZ t hif1 2S[q] q̇ V(q(t))dt2tiwhere q(t) is a possible classical trajectory, or path. According to Hamilton’s principle, the actual classical path is the one which extremises S –this gives Lagrange’s equations.The quantum amplitude for the particle to be at qf at time tf given thatit was at qi at time ti isM hqf e iĤ(tf ti )/h̄ qi i .According to Feynman, this amplitude is equivalently given by the pathintegralZI [dq] eiS[q]/h̄which is a integral over all functions (or paths) q(t) which satisfy q(ti ) qi ,q(tf ) qf . Obviously this needs to be better defined, but we will try tomake sense of it as we go along.

QFT15t2t1Figure 1: We can imagine doing the path integral by first fixing the values of q(t) at times(t1 , t2 , . . .).In order to understand why this might be true, first split the interval (ti , tf )into smaller pieces(tf , tn 1 , . . . , tj 1 , tj , . . . , t1 , ti )with tj 1 tj t. Our matrix element can then be writtenzM hqf eN factors} { iĤ t/h̄ iĤ t/h̄.e qi i(Note that we could equally well have considered a time-dependent hamiltonian, in which case each factor would be different.) Now insert a completeset of eigenstates of q̂ between each factor, eg at time-slice tj insertZ so thatM YZdq(tj ) q(tj )ihq(tj ) dq(tj )hq(tj 1 ) e iĤ t/h̄ q(tj )ijROn the other hand, we can think of doing the path integral [dq] by firstfixing the values {q(tj )} at times {tj } (see Fig. 1) and doing the integralsover the intermediate points on the path, and then doing the integral overthe {q(tj )}. ThusI YZZdq(tj )[dq(t)] e(i/h̄)R tj 1tj( 12 q̇2 V (q(t)))dtjThus we can prove that M I in general if we can show thatZhq(tj 1 ) e iĤ t/h̄ q(tj )i [dq(t)] e(i/h̄)R tj 1tj( 12 q̇2 V (q(t)))dt

QFT16for an arbitrarily short time interval t. First consider the case whenV 0. The path integral isZ[dq]e(i/2h̄)R tj 1tjq̇ 2 dtLet q(t) qc (t) δq(t) where qc (t) interpolates linearly between q(tj ) andq(tj 1 ), that isqc (t) q(tj ) ( t) 1 (t tj )(q(tj 1 ) q(tj ))and δq(tj 1 ) δq(tj ) 0. Then Z tj 1 2Zq(tj 1 ) q(tj ) 2 q̇ dt ( t) (δ q̇)2 dt ttjandZ[dq]e(i/2h̄)R tj 1tjq̇ 2 dt2 ei(q(tj 1 ) q(tj )) /2h̄ tZ[d(δq)]eR(i/2h̄) (δ q̇)2 dtThe second factor depends on t but not q(tj 1 ) or q(tj ), and can beabsorbed into the definition, or normalisation, of the functional integral.The first factor we recognise as the spreading of a wave packet initiallylocalised at q(tj ) over the time interval t. This is given by usual quantummechanics as2hq(tj 1 ) e ip̂ t/2h̄ q(tj )i(and this can be checked explicitly using the Schrödinger equation.)Now we argue, for V 6 0, that if t is small the spreading of the wavepacket is small, and therefore we can approximate V (q) by (say) V (q(tj )).Thus, as t 0,Z[dq]e(i/h̄)R tj 1tj( 12 q̇2 V (q(t)))dt hq(tj 1 ) e i( t/h̄)( 2 q̂1 2 V (q̂)) q(tj )iPutting all the pieces together, an integrating over the {q(tj )}, we obtainthe result we want.As well as being very useful for all sorts of computations, the path integralalso provides an intuitive way of thinking about classical mechanics as aRlimit of quantum mechanics. As h̄ 0 in the path integral [dq]eiS[q]/h̄ ,the important paths are those corresponding to stationary phase, whereδS[q]/δq 0. Other paths giving rapidly oscillating contributions andtherefore are suppressed. This is just Hamilton’s principle. In the semiclassical limit, the important paths will be those close to the classical one.

QFT17Periodic classical orbits will carry a complex phase which will in generalaverage to zero over many orbits. However if the action of a single orbitis 2πh̄ integer, the phase factor is unity and therefore such orbits willdominate the path integral. This is the Bohr-Sommerfeld quantisationcondition.The path integral is not restricted to hamiltonians of the above form, but ismore general. An important case is when Ĥ(â, ↠) is expressed in terms ofannihilation and creation operators â and ↠satisfying [â, ↠] 1. In thiscase, the path integral is obtained by replacing these by complex-valuedfunctions a(t) and a (t):Z [da][da ]e(i/h̄)R(ih̄a t a H(a,a ))dtThis is called a coherent state path integral. Similar versions exist forhamiltonians depending on quantum spins.2.1Imaginary time path integrals and statistical mechanicsSometimes it is useful to consider matrix elements of the formM hqf e Ĥ(τf τi )/h̄ qi i ,(1)that is, without the i. An analogous argument to the above shows thatthis is given by the path integralZwhereSE [q] [dq]e SE [q]/h̄Z τfτi(2)( 12 q̇ 2 V (q(τ )))dτThis is called the ‘imaginary time’ path integral: if we formally let t iτin the previous result, we get this answer. For reasons that will becomeapparent in the field theory generalisation, SE is usually referred to as theeuclidean action. Note that the relative sign of the kinetic and potentialterms changes between S and SE .One application of this idea is to quantum statistical mechanics. Thecanonical partition function in general isZ Tr e β Ĥ

QFT18where β 1/kB T . For the model under consideration the trace can bewrittenZZ dqi hqi e β Ĥ qi iwhere the matrix element is of the form (1) with τf τi βh̄. Thus Zis also given by the imaginary time path integral (2) over periodic pathssatisfying q(τi βh̄) q(τi ).Another application is to the computation of the ground state energy E0 .If we insert a complete set of eigenstates of Ĥ into (1) in the limit τf τi T , the leading term has the form e E0 T . On the other hand,in (2) this is given by paths q(τ ) which minimise SE [q]. Typically theymust satisfy q̇(τ ) 0 as τ . In most cases these have q̇ 0throughout, but other cases are more interesting. In particular this leadsto an understanding of quantum-mechanical tunnelling.The imaginary time path integral (2) may also be though of as a partitionfunction in classical statistical mechanics. Suppose that we treat τ as aspatial coordinate, and q(τ ) as the transverse displacement of a stretchedelastic string tethered at the points τi and τf . In addition a force, describedby an external potential V (q), acts on the string. The euclidean actionZSE [q] ( 12 m(dq/dτ )2 V (q(τ )))dτ(where we have restored the particle mass m in the original problem) cannow be thought of as the potential energy of the string, the first term representing the bending energy where m is the string tension. The partitionfunction of the string in classical statistical mechanics isZZ [dq][dp]e (R)/kB T1 22ρ p dτ SE [q]R1 2p dτ is the kineticwhere p now means the momentum density and 2ρenergy, with ρ being the string’s mass per unit length. The integral overp just gives a constant, as in a classical gas, so comparing with (2) we seethat the imaginary time path integral actually corresponds to a classicalpartition function at temperature kB T h̄. This is the simplest exampleof one of the most powerful ideas of theoretical physics: Quantum mechanics (in imaginary time) classical statisticalmechanics in one higher spatial dimension

QFT139Path integrals in field theoryA field theory is a system whose degrees of freedom are distributed throughout space. Since the continuous version of this is a little difficult to graspinitially, consider a discrete regular lattice in D-dimensional space whosesites are labelled by (x1 , x2 , x3 , . . .). At each site there is a degree of freedom. Instead of q̂ and p̂ we use φ̂ and π̂. Thusq̂ (φ̂(x1 ), φ̂(x2 ), . . . )p̂ (π̂(x1 ), π̂(x2 ), . . . )satisfying the canonical commutation relations[φ̂(xj ), π̂(xj 0 )] ih̄δjj 0The simplest form of the hamiltonian, generalising our single degree offreedom example, isĤ Xjĥ(π̂(xj ), φ̂(xj )) 12 JX(φ̂(xj ) φ̂(xj 0 ))2(jj 0 )where the last term couples the degrees of freedom on neighbouring sites.We can take ĥ to have the same form as before,ĥ(π̂(xj ), φ̂(xj )) 12 π̂(xj )2 V (φ̂(xj ))In the path integral version the operators φ̂(xj ) are replaced by c-numbervariables φ(xj , t):Z Y[dφ(xj , t)] e(i/h̄)S[{φ(xj ,t)}]jwhereZS X 1 ( 2 φ̇(xj , t)2j V (φ(xj , t))) 12 JX dt(φ(xj , t) φ(xj 0 , t))2 (jj 0 )This is the action for a lattice field theory.However we are interested in the continuum limit, as the lattice spacinga 0. The naive continuum limit is obtained by replacing sums overlattice sites by integrals:Z dD xX aDj

QFT110and making a gradient (Taylor) expansion of finite differences:X(φ(xj , ) φ̂(xj 0 , t))2 Z(jj 0 )dD x 2a ( φ(x, t))2DaAfter rescaling φ J 1/2 a(D 2)/2 φ (and also t), the action becomesZS dtdD x³1 22 φ̇ 12 ( φ)2 V (φ)This is the action for a classical field theory. The quantum theory is givenby the path integral over fields φ(x, t)Z[dφ(x, t)] eiS[φ]/h̄However, this begs the question of whether this has a meaningful limit asa 0. The naive answer is no, and making sense of this limit requires theunderstanding of renormalisation.3.1Field theory action functionalsThe example that we discussed above has several nice properties:R it is local : this means that S can be written as L(φ, φ̇, φ)dtdD xwhere the lagrangian density depends on the local value of the field andits derivatives. Moreover (more technically) it depends on derivativesonly up to second order. It can be shown that higher order derivativesin t lead to violations of causality. it is relativistically invariant (with c 1): in 4-vector (or D 1-vector)notation L can be writtenL 12 ( 0 φ)2 1X( i φ)22i V (φ) 12 ( µ φ)( µ φ) V (φ)so that if φ transforms as a Lorentz scalar, L is Lorentz invariant.This is of course a requirement for a field theory describing relativisticparticles. Another example isL 14 Fµν F µνwhere Fµν µ Aν ν Aµ and Aµ is a Lorentz vector. This is thelagrangian for the electromagnetic field. However in condensed matter

QFT111physics applications, relativistic invariance is not necessary (althoughit sometimes emerges anyway, with c replaced by the Fermi velocityor the speed of sound.) Note also that the imaginary time version ofthe action for our scalar field theory isZSE dX 1( i φ)22i 1 V (φ) dd xwhere d D 1 and dd x dD xdτ . That is, τ plays the same roleas a spatial coordinate and the theory is invariant under rotations ind-dimensional euclidean space. For this reason the imaginary timeversions are called euclidean quantum field theories. L should be invariant under any internal symmetries of the theory. Ifthis is φ φ, for example, then V should be an even function. Inthe case of electromagnetism, the symmetry is local gauge invariance. the theory be renormalisable (see later – although non-renormalisabletheories also play a role nowadays.)3.2The generating functionalOne difference between single particle quantum mechanics and quantumfield theory is that we are not usually interested in transition amplitudesbetween eigenstates φ(x)i of the field itself, as the field itself is not physically measurable. In fact, since we usually consider the limit of infinitespace, on relativistic grounds we should also consider infinite times. Thusthe only meaningful path integral would seem to beZ[dφ]e(i/h̄)R Rdt LdD x(3)which is just a number. In fact, if we consider the euclidean version of this,Z[dφ]e (1/h̄)R dτRLdD x(4)and relate this to a matrix element between eigenstates ni of Ĥ, we getlimX E (τ τ )n fiτf τi nehn ni e E0 (τf τi ) h0 0iThus we see that (4) (and, by careful definition through analytic continuation, see later, (3)) just tells about the vacuum vacuum amplitude,and is thus not very interesting (at least in flat space.)

QFT112In order to get any interesting physics we have to ‘tickle’ the vacuum, byadding sources which can make things happen. The simplest and mostuseful way of doing this is to add a source coupling locally to the fielditself, that is change the action toZS S J(x)φ(x)dd xThe vacuum amplitude is now a functional of this source function J(x):ZRZ[J] [dφ]eiS iJ(x)φ(x)dd xWe are now using x (x, t) to represent a point in Minkowski space (or(x, τ ) in euclidean space), and we have started using units where h̄ 1,both standard conventions in QFT. It is straightforward to put the rightfactors back in when we calculate a physical quantity.Since the i makes this rather ill-defined, we shall, for the time being, develop the theory in the euclidean versionZRZ[J] [dφ]e S J(x)φ(x)dd xInteresting physical quantities are found by taking functional derivativesof Z[J] with respect to J. For example 1 δZ[J] 1 Z[dφ]φ(x1 ) e S[φ] Z[0] δJ(x1 ) J 0 Z[0]By analogy with statistical mechanics in d dimensions, this can be thoughtof as an expectation value hφ(x1 )i. Similarly 1δ 2 Z[J]1 Z [dφ]φ(x1 )φ(x2 ) e S[φ] hφ(x1 )φ(x2 )i , Z[0] δJ(x1 )δJ(x2 ) J 0 Z[0]a correlation function.But what do these mean in the operator formulation? To see this imagineinserting a complete set of eigenstates. Then as τi and τf ,Z[dφ]φ(x1 ) e S[φ] e E0 (τf τi ) h0 φ̂(x1 ) 0iand the first factor gets cancelled by Z[0]. Similarly the two-point functionishφ(x1 )φ(x2 )i h0 φ̂(x1 )e (Ĥ E0 )(τ1 τ2 ) φ̂(x2 ) 0i

QFT113where we have emphasised that φ̂, in the Schrödinger picture, depends onthe spatial coordinates x but not τ . However if we go to the Heisenbergpicture and defineφ̂(x) e (Ĥ E0 )τ φ̂(x) e(Ĥ E0 )τthe rhs becomesh0 φ̂(x1 )φ̂(x2 ) 0i .However this is correct only if τ1 τ2 . If the inequality were reversed wewould have had to write the factors in the reverse order. Thus we concludethat · 1δ 2 Z[J] hφ(x1 )φ(x2 )i h0 T φ̂(x1 )φ̂(x2 ) 0i Z[0] δJ(x1 )δJ(x2 ) J 0where T arranges the operators in order of decreasing τ . Functional derivatives of Z[J] give vacuum expectation values oftime-ordered products of field operators This result continues to hold when we go back to real time t. Fortunatelyit is precisely these vacuum expectation values of time-ordered productswhich arise when we do scattering theory.In field theory, the correlation functions are also called Green functions(as we’ll see, for a free field theory they are Green functions of differentialoperators), or simply the N -point functions δ N Z[J]1 G(N ) (x1 , . . . , xN ) hφ(x1 ) . . . φ(xN )i Z[0] δJ(x1 ) . . . δJ(xN ) J 0EquivalentlyZ 1 ZXZ[J] dd x1 . . . dd xN G(N ) (x1 , . . . , xN )J(x1 ) . . . J(xN )Z[0] N 0 N !Z[J] is called the generating function for the N -point functions.It is also useful to defineW [J] log Z[J] ,which is analogous to the free energy in statis

Introduction to Quantum Field Theory John Cardy Michaelmas Term 2010 { Version 13/9/10 Abstract These notes are intendedtosupplementthe lecturecourse ‘Introduction toQuan-tum Field Theory’ and are not intended for wider distribution. Any errors or obvious omissions should be communicated to me at Contents 1 A Brief History of Quantum Field Theory 2 2 The Feynman .

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