Advanced Quantum field Theory lecture notes

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Preprint typeset in JHEP style - HYPER VERSION Michaelmas Term, 2006 and 2007 Quantum Field Theory University of Cambridge Part III Mathematical Tripos Dr David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OWA, UK 1 Recommended Books and Resources  M. Peskin and D. Schroeder, An Introduction to Quantum Field Theory This is a very clear and comprehensive book, covering everything in this course at the right level. It will also cover everything in the \Advanced Quantum Field Theory" course, much of the \Standard Model" course, and will serve you well if you go on to do research. To a large extent, our course will follow the rst section of this book. There is a vast array of further Quantum Field Theory texts, many of them with redeeming features. Here I mention a few very di erent ones.  S. Weinberg, The Quantum Theory of Fields, Vol 1 This is the rst in a three volume series by one of the masters of quantum eld theory. It takes a unique route to through the subject, focussing initially on particles rather than elds. The second volume covers material lectured in \AQFT".  L. Ryder, Quantum Field Theory This elementary text has a nice discussion of much of the material in this course.  A. Zee, Quantum Field Theory in a Nutshell This is charming book, where emphasis is placed on physical understanding and the author isn't afraid to hide the ugly truth when necessary. It contains many gems.  M Srednicki, Quantum Field Theory A very clear and well written introduction to the subject. Both this book and Zee's focus on the path integral approach, rather than canonical quantization that we develop in this course. There are also resources available on the web. Some particularly good ones are listed on the course webpage: 0. Introduction 1 0.1 Units and Scales 4 1. Classical Field Theory 7 1.1 The Dynamics of Fields 7 1.1.1 An Example: The Klein-Gordon Equation 8 1.1.2 Another Example: First Order Lagrangians 9 1.1.3 A Final Example: Maxwell's Equations 10 1.1.4 Locality, Locality, Locality 10 1.2 Lorentz Invariance 11 1.3 Symmetries 13 1.3.1 Noether's Theorem 13 1.3.2 An Example: Translations and the Energy-Momentum Tensor 14 1.3.3 Another Example: Lorentz Transformations and Angular Mo- mentum 16 1.3.4 Internal Symmetries 18 1.4 The Hamiltonian Formalism 19 2. Free Fields 21 2.1 Canonical Quantization 21 2.1.1 The Simple Harmonic Oscillator 22 2.2 The Free Scalar Field 23 2.3 The Vacuum 25 2.3.1 The Cosmological Constant 26 2.3.2 The Casimir E ect 27 2.4 Particles 29 2.4.1 Relativistic Normalization 31 2.5 Complex Scalar Fields 33 2.6 The Heisenberg Picture 35 2.6.1 Causality 36 2.7 Propagators 38 2.7.1 The Feynman Propagator 38 2.7.2 Green's Functions 40 2.8 Non-Relativistic Fields 41 2.8.1 Recovering Quantum Mechanics 43 1 3. Interacting Fields 47 3.1 The Interaction Picture 50 3.1.1 Dyson's Formula 51 3.2 A First Look at Scattering 53 3.2.1 An Example: Meson Decay 55 3.3 Wick's Theorem 56 3.3.1 An Example: Recovering the Propagator 56 3.3.2 Wick's Theorem 58 3.3.3 An Example: Nucleon Scattering 58 3.4 Feynman Diagrams 60 3.4.1 Feynman Rules 61 3.5 Examples of Scattering Amplitudes 62 3.5.1 Mandelstam Variables 66 3.5.2 The Yukawa Potential 67 4 3.5.3  Theory 69 3.5.4 Connected Diagrams and Amputated Diagrams 70 3.6 What We Measure: Cross Sections and Decay Rates 71 3.6.1 Fermi's Golden Rule 71 3.6.2 Decay Rates 73 3.6.3 Cross Sections 74 3.7 Green's Functions 75 3.7.1 Connected Diagrams and Vacuum Bubbles 77 3.7.2 From Green's Functions to S-Matrices 79 4. The Dirac Equation 81 4.1 The Spinor Representation 83 4.1.1 Spinors 85 4.2 Constructing an Action 87 4.3 The Dirac Equation 90 4.4 Chiral Spinors 91 4.4.1 The Weyl Equation 91 5 4.4.2 93 4.4.3 Parity 94 4.4.4 Chiral Interactions 95 4.5 Majorana Fermions 96 4.6 Symmetries and Conserved Currents 98 4.7 Plane Wave Solutions 100 4.7.1 Some Examples 102 2 4.7.2 Helicity 103 4.7.3 Some Useful Formulae: Inner and Outer Products 103 5. Quantizing the Dirac Field 106 5.1 A Glimpse at the Spin-Statistics Theorem 106 5.1.1 The Hamiltonian 107 5.2 Fermionic Quantization 109 5.2.1 Fermi-Dirac Statistics 110 5.3 Dirac's Hole Interpretation 110 5.4 Propagators 112 5.5 The Feynman Propagator 114 5.6 Yukawa Theory 115 5.6.1 An Example: Putting Spin on Nucleon Scattering 115 5.7 Feynman Rules for Fermions 117 5.7.1 Examples 118 5.7.2 The Yukawa Potential Revisited 121 5.7.3 Pseudo-Scalar Coupling 122 6. Quantum Electrodynamics 124 6.1 Maxwell's Equations 124 6.1.1 Gauge Symmetry 125 6.2 The Quantization of the Electromagnetic Field 128 6.2.1 Coulomb Gauge 128 6.2.2 Lorentz Gauge 131 6.3 Coupling to Matter 136 6.3.1 Coupling to Fermions 136 6.3.2 Coupling to Scalars 138 6.4 QED 139 6.4.1 Naive Feynman Rules 141 6.5 Feynman Rules 143 6.5.1 Charged Scalars 144 6.6 Scattering in QED 144 6.6.1 The Coulomb Potential 147 6.7 Afterword 149 3 Acknowledgements These lecture notes are far from original. My primary contribution has been to borrow, steal and assimilate the best discussions and explanations I could nd from the vast literature on the subject. I inherited the course from Nick Manton, whose notes form the backbone of the lectures. I have also relied heavily on the sources listed at the beginning, most notably the book by Peskin and Schroeder. In several places, for example the discussion of scalar Yukawa theory, I followed the lectures of Sidney Coleman, using the notes written by Brian Hill and a beautiful abridged version of these notes due to Michael Luke. My thanks to the many who helped in various ways during the preparation of this course, including Joe Conlon, Nick Dorey, Marie Ericsson, Eyo Ita, Ian Drummond, Jerome Gauntlett, Matt Headrick, Ron Horgan, Nick Manton, Hugh Osborn and Jenni Smillie. My thanks also to the students for their sharp questions and sharp eyes in spotting typos. I am supported by the Royal Society. 4 0. Introduction \There are no real one-particle systems in nature, not even few-particle systems. The existence of virtual pairs and of pair uctuations shows that the days of xed particle numbers are over." Viki Weisskopf The concept of wave-particle duality tells us that the properties of electrons and photons are fundamentally very similar. Despite obvious di erences in their mass and charge, under the right circumstances both su er wave-like di raction and both can pack a particle-like punch. Yet the appearance of these objects in classical physics is very di erent. Electrons and other matter particles are postulated to be elementary constituents of Nature. In contrast, light is a derived concept: it arises as a ripple of the electromagnetic eld. If photons and particles are truely to be placed on equal footing, how should we reconcile this di erence in the quantum world? Should we view the particle as fundamental, with the electromagnetic eld arising only in some classical limit from a collection of quantum photons? Or should we instead view the eld as fundamental, with the photon appearing only when we correctly treat the eld in a manner consistent with quantum theory? And, if this latter view is correct, should we also introduce an \electron eld", whose ripples give rise to particles with mass and charge? But why then didn't Faraday, Maxwell and other classical physicists nd it useful to introduce the concept of matter elds, analogous to the electromagnetic eld? The purpose of this course is to answer these questions. We shall see that the second viewpoint above is the most useful: the eld is primary and particles are derived concepts, appearing only after quantization. We will show how photons arise from the quantization of the electromagnetic eld and how massive, charged particles such as electrons arise from the quantization of matter elds. We will learn that in order to describe the fundamental laws of Nature, we must not only introduce electron elds, but also quark elds, neutrino elds, gluon elds, W and Z-boson elds, Higgs elds and a whole slew of others. There is a eld associated to each type of fundamental particle that appears in Nature. Why Quantum Field Theory? In classical physics, the primary reason for introducing the concept of the eld is to construct laws of Nature that are local. The old laws of Coulomb and Newton involve \action at a distance". This means that the force felt by an electron (or planet) changes 1 immediately if a distant proton (or star) moves. This situation is philosophically un- satisfactory. More importantly, it is also experimentally wrong. The eld theories of Maxwell and Einstein remedy the situation, with all interactions mediated in a local fashion by the eld. The requirement of locality remains a strong motivation for studying eld theories in the quantum world. However, there are further reasons for treating the quantum 1 eld as fundamental . Here I'll give two answers to the question: Why quantum eld theory? Answer 1: Because the combination of quantum mechanics and special relativity implies that particle number is not conserved. Particles are not indestructible objects, made at the beginning of the universe and here for good. They can be created and destroyed. They are, in fact, mostly ephemeral and eeting. This experimentally veri ed fact was rst predicted by Dirac who understood how relativity implies the necessity of anti-particles. An extreme demonstra- tion of particle creation is shown in the picture, which comes from the Relativistic Heavy Ion Collider (RHIC) at Brookhaven, Long Island. This machine crashes gold nu- clei together, each containing 197 nucleons. The resulting Figure 1: explosion contains up to 10,000 particles, captured here in all their beauty by the STAR detector. We will review Dirac's argument for anti-particles later in this course, together with the better understanding that we get from viewing particles in the framework of quan- tum eld theory. For now, we'll quickly sketch the circumstances in which we expect the number of particles to change. Consider a particle of mass m trapped in a box of size L. Heisenberg tells us that the uncertainty in the momentum is p =L. In a relativistic setting, momentum and energy are on an equivalent footing, so we should also have an uncertainty in the energy of order E c=L. However, when 2 the uncertainty in the energy exceeds E = 2mc , then we cross the barrier to pop particle anti-particle pairs out of the vacuum. We learn that particle-anti-particle pairs are expected to be important when a particle of mass m is localized within a distance of order  = mc 1 A concise review of the underlying principles and major successes of quantum eld theory can be found in the article by Frank Wilczek, 2 At distances shorter than this, there is a high probability that we will detect particle- anti-particle pairs swarming around the original particle that we put in. The distance is called the Compton wavelength. It is always smaller than the de Broglie wavelength  =h=jpj. If you like, the de Broglie wavelength is the distance at which the wavelike dB nature of particles becomes apparent; the Compton wavelength is the distance at which the concept of a single pointlike particle breaks down completely. The presence of a multitude of particles and antiparticles at short distances tells us that any attempt to write down a relativistic version of the one-particle Schr odinger equation (or, indeed, an equation for any xed number of particles) is doomed to failure. There is no mechanism in standard non-relativistic quantum mechanics to deal with changes in the particle number. Indeed, any attempt to naively construct a relativistic version of the one-particle Schr odinger equation meets with serious problems. (Negative probabilities, in nite towers of negative energy states, or a breakdown in causality are the common issues that arise). In each case, this failure is telling us that once we enter the relativistic regime we need a new formalism in order to treat states with an unspeci ed number of particles. This formalism is quantum eld theory (QFT). Answer 2: Because all particles of the same type are the same This sound rather dumb. But it's not What I mean by this is that two electrons are identical in every way, regardless of where they came from and what they've been through. The same is true of every other fundamental particle. Let me illustrate this through a rather prosaic story. Suppose we capture a proton from a cosmic ray which we identify as coming from a supernova lying 8 billion lightyears away. We compare this proton with one freshly minted in a particle accelerator here on Earth. And the two are exactly the same How is this possible? Why aren't there errors in proton production? How can two objects, manufactured so far apart in space and time, be identical in all respects? One explanation that might be o ered is that there's a sea of proton \stu " lling the universe and when we make a proton we somehow dip our hand into this stu and from it mould a proton. Then it's not surprising that protons produced in di erent parts of the universe are identical: they're made of the same stu . It turns out that this is roughly what happens. The \stu " is the proton eld or, if you look closely enough, the quark eld. In fact, there's more to this tale. Being the \same" in the quantum world is not like being the \same" in the classical world: quantum particles that are the same are truely indistinguishable. Swapping two particles around leaves the state completely unchanged apart from a possible minus sign. This minus sign determines the statis- tics of the particle. In quantum mechanics you have to put these statistics in by hand 3 and, to agree with experiment, should choose Bose statistics (no minus sign) for integer spin particles, and Fermi statistics (yes minus sign) for half-integer spin particles. In quantum eld theory, this relationship between spin and statistics is not something that you have to put in by hand. Rather, it is a consequence of the framework. What is Quantum Field Theory? Having told you why QFT is necessary, I should really tell you what it is. The clue is in the name: it is the quantization of a classical eld, the most familiar example of which is the electromagnetic eld. In standard quantum mechanics, we're taught to take the classical degrees of freedom and promote them to operators acting on a Hilbert space. The rules for quantizing a eld are no di erent. Thus the basic degrees of freedom in quantum eld theory are operator valued functions of space and time. This means that we are dealing with an in nite number of degrees of freedom at least one for every point in space. This in nity will come back to bite on several occasions. It will turn out that the possible interactions in quantum eld theory are governed by a few basic principles: locality, symmetry and renormalization group ow (the decoupling of short distance phenomena from physics at larger scales). These ideas make QFT a very robust framework: given a set of elds there is very often an almost unique way to couple them together. What is Quantum Field Theory Good For? The answer is: almost everything. As I have stressed above, for any relativistic system it is a necessity. But it is also a very useful tool in non-relativistic systems with many particles. Quantum eld theory has had a major impact in condensed matter, high- energy physics, cosmology, quantum gravity and pure mathematics. It is literally the language in which the laws of Nature are written. 0.1 Units and Scales Nature presents us with three fundamental dimensionful constants; the speed of lightc, Planck's constant (divided by 2) and Newton's constant G. They have dimensions 1 c = LT 2 1 = L MT 3 1 2 G = L M T Throughout this course we will work with \natural" units, de ned by c = = 1 (0.1) 4 which allows us to express all dimensionful quantities in terms of a single scale which 2 we choose to be mass or, equivalently, energy (since E = mc has become E = m). 9 The usual choice of energy unit iseV , the electron volt or, more oftenGeV = 10 eV or 12 TeV = 10 eV . To convert the unit of energy back to a unit of length or time, we need to insert the relevant powers of c and . For example, the length scale  associated to a mass m is the Compton wavelength  = mc 6 With this conversion factor, the electron mass m = 10 eV translates to a length scale e 12  = 2 10 m. e Throughout this course we will refer to the dimension of a quantity, meaning the d mass dimension. If X has dimensions of (mass) we will write X =d. In particular, the surviving natural quantity G has dimensions G =2 and de nes a mass scale, c 1 G = = (0.2) 2 2 M M p p 19 33 whereM  10 GeV is the Planck scale. It corresponds to a lengthl  10 cm. The p p Planck scale is thought to be the smallest length scale that makes sense: beyond this quantum gravity e ects become important and it's no longer clear that the concept of spacetime makes sense. The largest length scale we can talk of is the size of the 60 cosmological horizon, roughly 10 l . p Observable Planck Scale Cosmological Universe −33 Constant 20 billion light years 10 cm LHC Atoms Nuclei 10 −13 −8 Earth 10 cm 10 10 cm cm Energy length −3 −33 11 12 28 10 10 10 eV 10 −10 eV eV eV 19 = = 1 TeV 10 GeV Figure 2: Energy and Distance Scales in the Universe Some useful scales in the universe are shown in the gure. This is a logarithmic plot, with energy increasing to the right and, correspondingly, length increasing to the left. The smallest and largest scales known are shown on the gure, together with other relevant energy scales. The standard model of particle physics is expected to hold up 5 to about the TeV . This is precisely the regime that is currently being probed by the Large Hadron Collider (LHC) at CERN. There is a general belief that the framework of quantum eld theory will continue to hold to energy scales only slightly below the Planck scale for example, there are experimental hints that the coupling constants 18 of electromagnetism, and the weak and strong forces unify at around 10 GeV. For comparison, the rough masses of some elementary (and not so elementary) par- ticles are shown in the table, Particle Mass 2 neutrinos  10 eV electron 0.5 MeV Muon 100 MeV Pions 140 MeV Proton, Neutron 1 GeV Tau 2 GeV W,Z Bosons 80-90 GeV Higgs Boson 125 GeV 6 1. Classical Field Theory In this rst section we will discuss various aspects of classical elds. We will cover only the bare minimum ground necessary before turning to the quantum theory, and will return to classical eld theory at several later stages in the course when we need to introduce new ideas. 1.1 The Dynamics of Fields A eld is a quantity de ned at every point of space and time (x;t). While classical particle mechanics deals with a nite number of generalized coordinates q (t), indexed a by a label a, in eld theory we are interested in the dynamics of elds  (x;t) (1.1) a where botha andx are considered as labels. Thus we are dealing with a system with an in nite number of degrees of freedom at least one for each point x in space. Notice that the concept of position has been relegated from a dynamical variable in particle mechanics to a mere label in eld theory. An Example: The Electromagnetic Field The most familiar examples of elds from classical physics are the electric and magnetic elds, E(x;t) and B(x;t). Both of these elds are spatial 3-vectors. In a more sophis- ticated treatement of electromagnetism, we derive these two 3-vectors from a single  4-component eldA (x;t) = (;A) where = 0; 1; 2; 3 shows that this eld is a vector in spacetime. The electric and magnetic elds are given by A E =r and B =rA (1.2) t which ensure that two of Maxwell's equations,rB = 0 and dB=dt =rE, hold immediately as identities. The Lagrangian The dynamics of the eld is governed by a Lagrangian which is a function of (x;t), _ (x;t) andr(x;t). In all the systems we study in this course, the Lagrangian is of the form, Z 3 L(t) = d xL( ;  ) (1.3) a  a 7 where the ocial name forL is the Lagrangian density, although everyone simply calls it the Lagrangian. The action is, Z Z Z t 2 3 4 S = dt d xL = d xL (1.4) t 1 Recall that in particle mechanics L depends on q and q_, but not q . In eld theory _  we similarly restrict to LagrangiansL depending on  and , and not . In principle, 2 3 there's nothing to stopL depending onr,r ,r , etc. However, with an eye to later Lorentz invariance, we will only consider Lagrangians depending onr and not higher derivatives. Also we will not consider Lagrangians with explicit dependence on  x ; all such dependence only comes through  and its derivatives. We can determine the equations of motion by the principle of least action. We vary the path, keeping the end points xed and require S = 0, Z   L L 4 S = d x  + (  ) a  a  (  ) a  a Z      L L L 4 = d x  +  (1.5)  a  a  (  ) (  ) a  a  a The last term is a total derivative and vanishes for any  (x;t) that decays at spatial a in nity and obeys  (x;t ) =  (x;t ) = 0. Requiring S = 0 for all such paths a 1 a 2 yields the Euler-Lagrange equations of motion for the elds  , a   L L = 0 (1.6)  (  )   a a 1.1.1 An Example: The Klein-Gordon Equation Consider the Lagrangian for a real scalar eld (x;t), 1  1 2 2 L =    m  (1.7)   2 2 1 1 2 2 1 2 2 _ =  (r) m  2 2 2 where we are using the Minkowski space metric +1 1   = = (1.8)  1 1 Comparing (1.7) to the usual expression for the LagrangianL =TV , we identify the kinetic energy of the eld as Z 3 2 1 _ T = d x  (1.9) 2 8 and the potential energy of the eld as Z 3 2 2 2 1 1 V = d x (r) + m  (1.10) 2 2 The rst term in this expression is called the gradient energy, while the phrase \poten- tial energy", or just \potential", is usually reserved for the last term. To determine the equations of motion arising from (1.7), we compute L L 2  _ =m  and =  (;r) (1.11)  ( )  The Euler-Lagrange equation is then 2 2  r  +m  = 0 (1.12) which we can write in relativistic form as  2  +m  = 0 (1.13)  This is the Klein-Gordon Equation. The Laplacian in Minkowski space is sometimes 2 denoted by. In this notation, the Klein-Gordon equation reads +m  = 0. An obvious generalization of the Klein-Gordon equation comes from considering the Lagrangian with arbitrary potential V (), V 1   L =  V () )  + = 0 (1.14)   2  1.1.2 Another Example: First Order Lagrangians We could also consider a Lagrangian that is linear in time derivatives, rather than quadratic. Take a complex scalar eld whose dynamics is de ned by the real La- grangian i ? ? ? ? _ _ L = ( )r r m (1.15) 2 ? We can determine the equations of motion by treating and as independent objects, so that L i L i L _ = m and = and =r (1.16) ? ? _? 2 2 r This gives us the equation of motion 2 i =r +m (1.17) t This looks very much like the Schr odinger equation. Except it isn't Or, at least, the interpretation of this equation is very di erent: the eld is a classical eld with none of the probability interpretation of the wavefunction. We'll come back to this point in Section 2.8. 9 The initial data required on a Cauchy surface di ers for the two examples above. 2 _ _ WhenL  , both  and  must be speci ed to determine the future evolution; ? ? _ however whenL , only and are needed. 1.1.3 A Final Example: Maxwell's Equations We may derive Maxwell's equations in the vacuum from the Lagrangian,    2 1 1 L = ( A ) ( A ) + ( A ) (1.18)    2 2 Notice the funny minus signs This is to ensure that the kinetic terms forA are positive i 1 2 _ using the Minkowski space metric (1.8), soL A . The Lagrangian (1.18) has no i 2 2 _ kinetic term A for A . We will see the consequences of this in Section 6. To see that 0 0 Maxwell's equations indeed follow from (1.18), we compute L     = A + ( A ) (1.19)  ( A )   from which we may derive the equations of motion,   L 2         = A + ( A ) = ( A A ) F (1.20)     ( A )   where the eld strength is de ned by F = A A . You can check using (1.2)      that this reproduces the remaining two Maxwell's equations in a vacuum: rE = 0 and E=t =rB. Using the notation of the eld strength, we may rewrite the Maxwell Lagrangian (up to an integration by parts) in the compact form  1 L = F F (1.21)  4 1.1.4 Locality, Locality, Locality In each of the examples above, the Lagrangian is local. This means that there are no terms in the Lagrangian coupling (x;t) directly to (y;t) with x =6 y. For example, there are no terms that look like Z 3 3 L = d xd y (x)(y) (1.22) A priori, there's no reason for this. After all, x is merely a label, and we're quite happy to couple other labels together (for example, the term A A in the Maxwell 3 0 0 3 Lagrangian couples the  = 0 eld to the  = 3 eld). But the closest we get for the 2 x label is a coupling between (x) and (x +x) through the gradient term (r) . This property of locality is, as far as we know, a key feature of all theories of Nature. Indeed, one of the main reasons for introducing eld theories in classical physics is to implement locality. In this course, we will only consider local Lagrangians. 10 1.2 Lorentz Invariance The laws of Nature are relativistic, and one of the main motivations to develop quantum eld theory is to reconcile quantum mechanics with special relativity. To this end, we want to construct eld theories in which space and time are placed on an equal footing and the theory is invariant under Lorentz transformations,  0    x (x ) =  x (1.23)   where  satis es         = (1.24)   3 1 For example, a rotation by about thex -axis, and a boost byv 1 along thex -axis are respectively described by the Lorentz transformations 0 1 0 1 1 0 0 0 v 0 0 B C B C B C B C 0 cos sin 0 v 0 0   B C B C  = and  = (1.25)   B C B C 0 sin cos 0 0 0 1 0 A A 0 0 0 1 0 0 0 1 p 2 with = 1= 1v . The Lorentz transformations form a Lie group under matrix multiplication. You'll learn more about this in the \Symmetries and Particle Physics" course. The Lorentz transformations have a representation on the elds. The simplest ex- ample is the scalar eld which, under the Lorentz transformation x x, transforms as 0 1 (x) (x) =( x) (1.26) 1 The inverse  appears in the argument because we are dealing with an active trans- formation in which the eld is truly shifted. To see why this means that the inverse appears, it will suce to consider a non-relativistic example such as a temperature eld. Suppose we start with an initial eld (x) which has a hotspot at, say, x = (1; 0; 0). 0 After a rotationxRx about the z-axis, the new eld  (x) will have the hotspot at 0 x = (0; 1; 0). If we want to express  (x) in terms of the old eld , we need to place ourselves atx = (0; 1; 0) and ask what the old eld looked like where we've come from 1 1 at R (0; 1; 0) = (1; 0; 0). This R is the origin of the inverse transformation. (If we were instead dealing with a passive transformation in which we relabel our choice of 0 coordinates, we would have instead (x) (x) =(x)). 11 The de nition of a Lorentz invariant theory is that if (x) solves the equations of 1 motion then ( x) also solves the equations of motion. We can ensure that this property holds by requiring that the action is Lorentz invariant. Let's look at our examples: Example 1: The Klein-Gordon Equation 0 1 For a real scalar eld we have (x)  (x) = ( x). The derivative of the scalar eld transforms as a vector, meaning 1  ( )(x) ( ) ( )(y)    1 where y =  x. This means that the derivative terms in the Lagrangian density transform as  1  1   L (x) = (x) (x) ( ) ( )(y) ( ) ( )(y)  deriv        = ( )(y) ( )(y)   = L (y) (1.27) deriv 2 2 The potential terms transform in the same way, with  (x)  (y). Putting this all together, we nd that the action is indeed invariant under Lorentz transformations, Z Z Z 4 4 4 S = d xL(x) d xL(y) = d yL(y) =S (1.28) where, in the last step, we need the fact that we don't pick up a Jacobian factor when R R 4 4 we change integration variables from d x to d y. This follows because det  = 1. (At least for Lorentz transformation connected to the identity which, for now, is all we deal with). Example 2: First Order Dynamics In the rst-order Lagrangian (1.15), space and time are not on the same footing. (L is linear in time derivatives, but quadratic in spatial derivatives). The theory is not Lorentz invariant. In practice, it's easy to see if the action is Lorentz invariant: just make sure all the Lorentz indices  = 0; 1; 2; 3 are contracted with Lorentz invariant objects, such as the metric  . Other Lorentz invariant objects you can use include the totally  antisymmetric tensor  and the matrices that we will introduce when we come   to discuss spinors in Section 4. 12 Example 3: Maxwell's Equations    1 Under a Lorentz transformationA (x)  A ( x). You can check that Maxwell's  Lagrangian (1.21) is indeed invariant. Of course, historically electrodynamics was the rst Lorentz invariant theory to be discovered: it was found even before the concept of Lorentz invariance. 1.3 Symmetries The role of symmetries in eld theory is possibly even more important than in particle mechanics. There are Lorentz symmetries, internal symmetries, gauge symmetries, supersymmetries.... We start here by recasting Noether's theorem in a eld theoretic framework. 1.3.1 Noether's Theorem  Every continuous symmetry of the Lagrangian gives rise to a conserved current j (x) such that the equations of motion imply  j = 0 (1.29)  0 or, in other words, j =t +rj = 0. A Comment: A conserved current implies a conserved charge Q, de ned as Z 3 0 Q = d x j (1.30) 3 R which one can immediately see by taking the time derivative, Z Z 0 dQ j 3 3 = d x = d xrj = 0 (1.31) dt 3 t 3 R R assuming that j 0 suciently quickly asjxj1. However, the existence of a current is a much stronger statement than the existence of a conserved charge because it implies that charge is conserved locally. To see this, we can de ne the charge in a nite volume V , Z 3 0 Q = d x j (1.32) V V Repeating the analysis above, we nd that Z Z dQ V 3 = d xrj = jdS (1.33) dt V A 13 where A is the area bounding V and we have used Stokes' theorem. This equation means that any charge leaving V must be accounted for by a ow of the current 3- vectorj out of the volume. This kind of local conservation of charge holds in any local eld theory. Proof of Noether's Theorem: We'll prove the theorem by working in nitesimally. We may always do this if we have a continuous symmetry. We say that the transfor- mation  (x) =X () (1.34) a a is a symmetry if the Lagrangian changes by a total derivative,  L = F (1.35)   for some set of functionsF (). To derive Noether's theorem, we rst consider making an arbitrary transformation of the elds  . Then a L L L =  + ( ) a  a  (  ) a  a     L L L =  +  (1.36)  a  a  (  ) (  ) a  a  a When the equations of motion are satis ed, the term in square brackets vanishes. So we're left with   L L =  (1.37)  a (  )  a  But for the symmetry transformation  =X (), we have by de nition L = F . a a  Equating this expression with (1.37) gives us the result L    j = 0 with j = X ()F () (1.38)  a (  )  a 1.3.2 An Example: Translations and the Energy-Momentum Tensor Recall that in classical particle mechanics, invariance under spatial translations gives rise to the conservation of momentum, while invariance under time translations is responsible for the conservation of energy. We will now see something similar in eld theories. Consider the in nitesimal translation     x x  )  (x) (x) +  (x) (1.39) a a  a 14

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