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Introduction to Quantum Field Theory

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Introduction to Quantum Field Theory John Cardy Michaelmas Term 2010 { Version 13/9/10 Abstract These notes are intended to supplement the lecture course `Introduction to Quan- tum Field Theory' and are not intended for wider distribution. Any errors or obvious omissions should be communicated to me at [email protected]. Contents 1 A Brief History of Quantum Field Theory 2 2 The Feynman path integral in particle quantum mechanics 4 2.1 Imaginary time path integrals and statistical mechanics . 7 3 Path integrals in ¯eld theory 9 3.1 Field theory action functionals . 10 3.2 The generating functional . 11 3.3 The propagator in free ¯eld theory . 14 4 Interacting ¯eld theories 18 4.1 Feynman diagrams . 18 4.2 Evaluation of Feynman diagrams . 26 5 Renormalisation 29 5.1 Analysis of divergences . 29 5.2 Mass, ¯eld, and coupling constant renormalisation . 32 1 QFT1 2 6 Renormalisation Group 39 6.1 Callan-Symanzik equation . 40 6.2 Renormalisation group flows . 41 6.3 One-loop computation in ¸Á4 theory . 44 6.4 Application to critical behaviour in statistical mechanics . 46 6.5 Large N ............................ 50 7 From Feynman diagrams to Cross-sections 53 7.1 The S-matrix: analyticity and unitarity . 58 8 Path integrals for fermions 62 1 A Brief History of Quantum Field Theory Quantum ¯eld theory (QFT) is a subject which has evolved considerably over the years and continues to do so. From its beginnings in elementary particle physics it has found applications in many other branches of science, in particular condensed matter physics but also as far a¯eld as biology and economics. In this course we shall be adopting an approach (the path integral) which was not the original one, but became popular, even essential, with new advances in the 1970s. However, to set this in its context, it is useful to have some historical perspective on the development of the subject (dates are only rough). ² 19th C. Maxwell's equations { a classical ¯eld theory for electromag- netism. ² 1900: Planck hypothesises the photon as the quantum of radiation. ² 1920s/30s: development of particle quantum mechanics: the same rules when applied to the Maxwell ¯eld predict photons. However relativistic particle quantum mechanics has problems (negative energy states.) ² 1930s/40s: realisation that relativity + quantum mechanics, in which particles can be created and destroyed, needs a many-particle descrip- QFT1 3 tion where the particles are the quanta of a quantised classical ¯eld theory, in analogy with photons. ² 1940s: formulation of the calculation rules for quantum electrodynam- ics (QED) { Feynman diagrams; the formulation of the path integral approach. ² 1950s: the understanding of how to deal with the divergences of Feyn- man diagrams through renormalisation; QFT methods begin to be applied to other many-body systems eg in condensed matter. ² 1960s: QFT languishes { how can it apply to weak + strong interac- tions? ² 1970s: renormalisation of non-Abelian gauge theories, the renormal- isation group (RG) and asymptotic freedom; the formulation of the Standard Model ² 1970s: further development of path integral + RG methods: applica- tions to critical behaviour. ² 1970s: non-perturbative methods, lattice gauge theory. ² 1980s: string theory + quantum gravity, conformal ¯eld theory (CFT); the realisation that all quantum ¯eld theories are only e®ective over some range of length and energy scales, and those used in particle physics are no more fundamental than in condensed matter. ² 1990s/2000s: holography and strong coupling results for gauge ¯eld theories; many applications of CFT in condensed matter physics. Where does this course ¯t 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 exper- imental particle physicists, through high energy theorists, to condensed matter physicists (with maybe a few theoretical chemists, quantum com- puting types and mathematicians thrown in). Therefore all I can hope to do is to give you some of the basic ideas, illustrated in their most simple contexts. The hope is to take you all from the Feynman path integral, through a solid grounding in Feynman diagrams, to renormalisation and QFT1 4 the RG. From there hopefully you will have enough background to un- derstand Feynman diagrams and their uses in particle physics, and have the basis for understanding gauge theories as well as applications of ¯eld theory and RG methods in condensed matter physics. 2 The Feynman path integral in particle quantum mechanics In this lecture we will recall the Feynman path integral for a system with a single degree of freedom, in preparation for the ¯eld theory case of many degrees of freedom. Consider a non-relativistic particle of unit mass moving in one dimension. The coordinate operator isq ^, and the momentum operator isp ^. (I'll be careful to distinguish operators and c-numbers.) Of course [^q; p^] = ih¹. We denote the eigenstates ofq ^ by jq0i, thusq ^jq0i = q0jq0i, and hq0jq00i = ±(q0 ¡ q00). ^ 1 2 Suppose the hamiltonian has the form H = 2p^ + V (^q) (we can consider more general forms { see later.) The classical action corresponding to this is Z t h i f 1 2 S[q] = 2q_ ¡ V (q(t)) dt ti where q(t) is a possible classical trajectory, or path. According to Hamil- ton'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 that it was at qi at time ti is ¡iH^ (tf ¡ti)=¹h M = hqf je jqii : According to Feynman, this amplitude is equivalently given by the path integral Z I = [dq] eiS[q]=¹h which is a integral over all functions (or paths) q(t) which satisfy q(ti) = qi, q(tf ) = qf . Obviously this needs to be better de¯ned, but we will try to make sense of it as we go along. QFT1 5 t2 t1 Figure 1: We can imagine doing the path integral by ¯rst ¯xing the values of q(t) at times (t1; t2;:::). In order to understand why this might be true, ¯rst 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 written N factors z }| { ¡iH^ ¢t=¹h ¡iH^ ¢t=¹h M = hqf j e : : : e jqii (Note that we could equally well have considered a time-dependent hamil- tonian, in which case each factor would be di®erent.) Now insert a complete set of eigenstates ofq ^ between each factor, eg at time-slice tj insert Z 1 dq(tj)jq(tj)ihq(tj)j ¡1 so that Z Y ¡iH^ ¢t=¹h M = dq(tj)hq(tj+1)je jq(tj)i j R On the other hand, we can think of doing the path integral [dq] by ¯rst ¯xing the values fq(tj)g at times ftjg (see Fig. 1) and doing the integrals over the intermediate points on the path, and then doing the integral over the fq(tj)g. Thus Z Z R t Y j+1 1 2 (i=¹h) t ( 2 q_ ¡V (q(t)))dt I = dq(tj) [dq(t)] e j j Thus we can prove that M = I in general if we can show that Z R t ^ j+1 1 2 ¡iH¢t=¹h (i=¹h) t ( 2 q_ ¡V (q(t)))dt hq(tj+1)je jq(tj)i = [dq(t)] e j QFT1 6 for an arbitrarily short time interval ¢t. First consider the case when V = 0. The path integral is Z R t (i=2¹h) j+1 q_2dt [dq]e tj Let q(t) = qc(t) + ±q(t) where qc(t) interpolates linearly between q(tj) and q(tj+1), that is ¡1 qc(t) = q(tj) + (¢t) (t ¡ tj)(q(tj+1) ¡ q(tj)) and ±q(tj+1) = ±q(tj) = 0. Then 0 1 Z 2 Z tj+1 2 q(tj+1) ¡ q(tj) 2 q_ dt = (¢t) @ A + (±q_) dt tj ¢t and Z R t 2 Z R j+1 2 2 (i=2¹h) t q_ dt i(q(t )¡q(t )) =2¹h¢t (i=2¹h) (±q_) dt [dq]e j = e j+1 j [d(±q)]e The second factor depends on ¢t but not q(tj+1) or q(tj), and can be absorbed into the de¯nition, or normalisation, of the functional integral. The ¯rst factor we recognise as the spreading of a wave packet initially localised at q(tj) over the time interval ¢t. This is given by usual quantum mechanics as ¡ip^2¢t=2¹h hq(tj+1)je jq(tj)i (and this can be checked explicitly using the SchrÄodingerequation.) Now we argue, for V 6= 0, that if ¢t is small the spreading of the wave packet is small, and therefore we can approximate V (q) by (say) V (q(tj)). Thus, as ¢t ! 0, Z R t j+1 1 2 1 2 (i=¹h) t ( 2 q_ ¡V (q(t)))dt ¡i(¢t=¹h)( q^ +V (^q)) [dq]e j » hq(tj+1)je 2 jq(tj)i Putting all the pieces together, an integrating over the fq(tj)g, we obtain the result we want. As well as being very useful for all sorts of computations, the path integral also provides an intuitive way of thinking about classical mechanics as a R limit of quantum mechanics. Ash ¹ ! 0 in the path integral [dq]eiS[q]=¹h, the important paths are those corresponding to stationary phase, where ±S[q]=±q = 0.
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